the son of Ulysses and Penelope. His character and adventures belong more to poetry than to history.

**TELESCOPE.**

*Telescope,* an optical instrument for viewing distant objects; so named by compounding the Greek words ῥάσις, *far off,* and ἐστία, *I look at or contemplate.* This name is commonly appropriated to the larger sizes of the instrument, while the smaller are called *perspective glasses,* *spy-glasses,* *opera-glasses.* A particular kind, which is thought to be much brighter than the rest, is called a *night-glass.*

The history of the invention of the telescope has been given in our article Optics. The theory of the Astronomical Telescope will be found in Astronomy (part iv. chap. 4), where two particular constructions of the instrument, viz. that of Professor Barlow and Mr Rogers, are described; and also the Dorpat telescope, of which a figure is there given. At the time that article was written, this was the most powerful instrument of the kind that had ever been directed to the heavens, its object-glass being 9½ inches in diameter, and focal length about 14 feet. Now, however, the University of Cambridge possesses an instrument still more powerful, the Northumberland Equatorial, so called in honour of the noble duke by whom it was presented to the university in the year 1838. The object-glass of this magnificent telescope is 11½ clear inches in diameter, and its focal length 19½ feet. A particular description of it has been given in the eleventh volume of the Cambridge Astronomical Observations. There is another telescope of still larger dimensions in the possession of an amateur astronomer, Mr Cooper of Sligo. The diameter of the object-glass is 13¾ inches, and the focal length 21 feet 3 inches.

The general aim in the construction of a telescope is, to form, by means of mirrors or lenses, an image of the distant object, as large, as bright, and as extensive as is possible, consistently with distinctness; and then to view the image with a magnifying glass in any convenient manner. This gives us an arrangement of our subject. We shall first show the principles of construction of the object-glass or mirror, so that it shall form an image of the distant object with these qualities; and then show how to construct the magnifying glass or eye-piece, so as to preserve them unimpaired.

This indistinctness which we wish to avoid arises from two causes; the spherical figures of the refracting and reflecting surfaces, and the different refrangibility of the differently coloured rays of light. The first may be called the *spherical* and the second the *chromatic* indistinctness; and the deviations from the foci may be called the *spherical* and the *chromatic* aberrations.

The limits of a work like this will not permit us to give any more of the doctrine of aberrations than is absolutely necessary for the construction of achromatic telescopes; and we must refer the reader for a general view of the whole to Euler's Dioptrics, also to Dr Smith's Optics. We shall begin with the aberration of colour.

Let white or compounded light fall perpendicularly on the flat side PQ (fig. 1) of a plano-convex lens PVQ, whose axis is CV and vertex V. The white ray PP falling on the extremity of the lens is dispersed by refraction at the point P of the spherical surface, and the red ray goes to the point r of the axis, and the violet ray to the point v. In like manner, the white ray QP is dispersed by refraction at Q, the red ray going to r, and the violet to v. The red ray PR crosses the violet ray Qv in a point D, and Qr crosses Pv in a point E; and the whole light refracted and dispersed by the circumference whose diameter is PQ, passes through the circular area, whose diameter is DE. Supposing that the lens is of such a form that it would collect red rays, refracted by its whole surface in the point r, and violet in the point v; then it is evident that the whole light which occupies the surface of the lens will pass through this little circle, whose diameter is DE. Therefore white light issuing from a point so distant that the rays may be considered as parallel, will not be collected.

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1 Though this subject has already been partly discussed under the articles Achromatic Glasses and Optics, it could not be viewed as fully illustrated without a separate treatise; and the editor gladly avails himself of the late Professor Robison's article on the Telescope, which forms one of the valuable series of contributions by him to the third edition of this work. in another point or focus, but will be dispersed over the surface of that little circle, which is therefore called the circle of chromatic dispersion; and the radiant point will be represented by this circle. The neighbouring points are in like manner represented by circles; and these circles, encroaching on and mixing with each other, must occasion haziness or confusion, and render the picture indistinct. This indistinctness will be greater in the proportion of the number of circles which are in this manner mixed together. This will be in the proportion of the room that is for them; that is, in proportion to the area of the circle, or in the duplicate proportion of its diameter. Our first business therefore is to obtain measures of this diameter, and to mark the connection between it and the aperture and focal distance of the lens.

Let \( i \) be to \( r \) as the sine of incidence in glass to the sine of refraction of the red rays; and let \( i \) be to \( v \) as the sine of incidence to the sine of refraction of the violet rays. Then we say, that when the aperture \( PQ \) is moderate, \( v - r : v + r - 2i = DE : PQ \) very nearly.

For let \( DE \), which is evidently perpendicular to \( Vr \), meet the parallel incident rays in \( K \) and \( L \) and the radii of the spherical surface in \( G \) and \( H \). It is plain that \( GPK \) is equal to the angle of incidence on the posterior or spherical surface of the lens; and \( GPv \) and \( GPv \) are the angles of the refraction of the red and the violet rays; and that \( GK \), \( GD \), and \( GE \) are very nearly as the sines of those angles, because the angles are supposed to be small. Therefore \( DE : KD = v - r : r - i \); and, by doubling the consequents, \( DE : 2KD = v - r : 2r - 2i \). Also \( DE : 2KD + DE = v - r : 2r - 2i + v - r = v - r : r + v - 2i \).

But \( 2KD + DE \) is equal to \( KL \) or \( PQ \). Therefore we have \( DE : PQ = v - r : r + v - 2i \). Q.E.D.

Cor. 1. Sir Isaac Newton found, that in common glass the sines of refraction of the red and violet rays were 77 and 78 where the sine of incidence was 50. Hence \( v - r \) is to \( v + r - 2i \) as 1 to 55; and the diameter of the smallest circle of dispersion is \( \frac{1}{3} \)th part of that of the lens.

2. In like manner may be determined the circle of dispersion that will comprehend the rays of any particular colour or set of colours. Thus all the orange and yellow will pass through a circle whose diameter is \( \frac{1}{3} \)th of that of the lens.

3. In different surfaces, or plano-convex lenses, the angles of aberration \( Pr \) are as the breadth \( PQ \) directly, and as the focal distance \( VF \) inversely; because any angle \( DPE \) is as its subtense \( DE \) directly and radius \( DP \) inversely. — We call \( VF \) the focal distance, because at this distance, or at the point \( F \), the light is most of all constipated. If we examine the focal distance by holding the lens to the sun, we judge it to be where the light is drawn into the smallest spot.

When we reflect that a lens of \( \frac{1}{3} \) inches in diameter has a circle of dispersion \( \frac{1}{3} \)th of an inch in diameter, we are surprised that it produces any picture of an object that can be distinguished. We should not expect greater distinctness from such a lens than would be produced in a camera obscura without a lens, by simply admitting the light through a hole of \( \frac{1}{3} \)th of an inch in diameter. This, we know, would be very hazy and confused. But when we remark the superior vivacity of the yellow and orange light in comparison with the rest, we may believe that the effect produced by the confusion of the other colours will be much less sensible. But a stronger reason is, that the light is much denser in the middle of the circle of dispersion, and is exceedingly faint towards the margin. This, however, must not be taken for granted; and we must know distinctly the manner in which the light of different colours is distributed over the circle of chromatic dispersion, before we pretend to pronounce on the immense difference between the indistinctness arising from colour and that arising from the spherical figure. We think this the more necessary, seeing that Telescope, the illustrious discoverer of the chromatic aberration has made a great mistake in the comparison, because he did not consider the distribution of the light in the circle of spherical dispersion. It is therefore proper to investigate the chromatic distribution of the light, and we shall then see that the superiority of the reflecting telescope is incomparably less than Newton imagined it to be.

Therefore let \( EB \) (fig. 2) represent a plano-convex lens, of which \( C \) is the centre and \( Cr \) the axis.

Let us suppose it to have no spherical aberration, but to collect rays occupying its whole surface to single points in the axis. Let a beam of white or compounded light fall perpendicularly on its plane surface. The rays will be so refracted by its curved surface, that the extreme red rays will be collected at \( r \), the extreme violet rays at \( v \), and those of intermediate refrangibility at intermediate points \( o, y, g, b, p, v \) of the line \( rv \), which is nearly \( \frac{1}{3} \)th of \( rC \). The extreme red and violet rays will cross each other at \( A \) and \( D \); and \( AD \) will be a section or diameter of the circle of chromatic dispersion, and will be about \( \frac{1}{3} \)th of \( EB \). We may suppose \( wr \) to be bisected in \( b \), because \( ab \) is to \( br \) very nearly in the ratio of equality (for \( rb : rC = bA : cE = bA : cB = wb : wC \)). The line \( rw \) will be a kind of prismatic spectrum, red from \( r \) to \( o \), orange-coloured from \( o \) to \( y \), yellow from \( y \) to \( g \), green from \( g \) to \( b \), blue from \( b \) to \( p \), purple from \( p \) to \( v \), and violet from \( v \) to \( w \).

The light in its compound state must be supposed uniformly dense as it falls upon the lens; and the same must be said of the rays of any particular colour. Newton supposes also, that when a white ray, such as \( eE \), is dispersed into its component coloured rays by refraction at \( E \), it is uniformly spread over the angle \( DEA \). This supposition is indeed gratuitous; but we have no argument to the contrary, and may therefore consider it as just. The consequence is, that each point \( w, v, p, b, \&c. \) of the spectrum is not only equally luminous, but also illuminates uniformly its corresponding portion of \( AD \); that is to say, the coating (so to term it) of any particular colour, such as purple, from the point \( p \), is uniformly dense in every part of \( AD \) on which it falls. In like manner, the colouring of yellow, intercepted by a part of \( AD \) in its passage to the point \( y \), is uniformly dense in all its parts. But the density of the different colours in \( AD \) is extremely different; for, since the radiation in \( w \) is equally dense with that in \( p \), the density of the violet colouring, which radiates from \( w \), and is spread over the whole of \( AD \), must be much less than the density of the purple colouring, which radiates from \( p \), and occupies only a part of \( AD \) round the circle \( b \). These densities must be very nearly in the inverse proportion of \( wb^2 : pb^2 \). Hence we see that the central point \( b \) will be very intensely illuminated by the blue radiating from \( pb \), and the green intercepted from \( bg \). It will be more faintly illuminated by the purple radiating from \( yp \), and the yellow intercepted from \( gy \); and still more faintly by the violet from \( sv \), and the orange and red intercepted from \( yr \). The whole colouring will be a white, tending a little to yellowness.

The accurate proportion of these colourings may be computed from our knowledge of the position of the points \( o \), \( y \), \( g \), &c. But this is of little moment. It is of more consequence to be able to determine the proportion of the total intensity of the light in \( b \) to its intensity in any other point \( I \).

For this purpose draw \( rIR \), \( lwW \), meeting the lens in \( R \) and \( W \). The point \( I \) receives none of the light which passes through the space \( RW \); for it is evident that \( bI : CR = bA : CE = 1 : 55 \), and that \( CR = CW \); and therefore, since all the light incident on \( EB \) passes through \( AD \), all the light incident on \( RW \) passes through \( Ib \) (\( bI \) being made \( = bI \)). Draw \( oIO \), \( yIN \), \( gIG \), \( lpP \), \( lwW \). It is plain that \( I \) receives red light from \( RO \), orange from \( OY \), yellow from \( YG \), green from \( GE \), a little blue from \( BP \), purple from \( PV \), and violet from \( VW \). It therefore wants some of the green and of the blue.

That we may judge of the intensity of these colours at \( I \), suppose the lens covered with paper pierced with a small hole at \( G \). The green light only will pass through \( I \); the other colours will pass between \( I \) and \( b \), or between \( I \) and \( A \), according as they are more or less refrangible than the particular green at \( I \). This particular colour converges to \( g \), and therefore will illuminate a small spot round \( I \), where it will be as much denser than it is at \( G \) as this spot is smaller than the hole at \( G \). The natural density at \( G \), therefore, will be to the increased density at \( I \), as \( gI^2 \) to \( gG^2 \), or as \( gI^2 \) to \( gC^2 \), or as \( bI^2 \) to \( CG^2 \). In like manner, the natural density of the purple coming to \( I \) through an equal hole at \( P \) will be to the increased density at \( I \), as \( bI^2 \) to \( CP^2 \). And thus it appears, that the intensity of the differently coloured illuminations of any point of the circle of dispersion, is inversely proportional to the square of the distance from the centre of the lens to the point of its surface through which the colouring light comes to this point of the circle of dispersion. This circumstance will give us a very easy, and, we think, an elegant solution of the question.

Bisect \( CE \) in \( F \), and draw \( FL \) perpendicular to \( CE \), making it equal to \( CF \). Through the point \( L \) describe the hyperbola \( KLN \) of the second order, that is, having the ordinates \( EK \), \( FL \), \( RN \), &c. inversely proportional to the squares of the abscisse \( CE \), \( CF \), \( CR \), &c.; so that \( FL : RN = \frac{1}{CF^2} : \frac{1}{CR^2} \), or \( CR^2 : CF^2 \), &c. It is evident that these ordinates are proportional to the densities of the severally coloured lights which go from them to any points whatever of the circle of dispersion.

Now the total density of the light at \( I \) depends both on the density of each particular colour and on the number of colours which fall on it. The ordinates of this hyperbola determine the first; and the space \( ER \) measures the number of colours which fall on \( I \), because it receives light from the whole of \( ER \), and of its equal \( BW \). Therefore, if ordinates be drawn from any point of \( ER \), their sum will be as the whole light which goes to \( I \); that is, the total density of the light at \( I \) will be proportional to the area \( NREK \). Now it is known that \( CE \cdot EK \) is equal to the infinitely extended area lying beyond \( EK \); and \( CR \cdot RN \) is equal to the infinitely extended area lying beyond \( RN \). Therefore the area \( NREK \) is equal to \( CR \cdot RN - CE \cdot EK \). But \( RN \) and \( EK \) are respectively equal to \( \frac{CF^3}{CR^2} \) and \( \frac{CF^3}{CE^2} \). Therefore

\[ \text{the density at } I \text{ is proportional to } \left( \frac{CR}{CF^2} - \frac{CE}{CF^2} \right) = \frac{CF^3}{CR^2} \cdot \frac{CE}{CR} = \frac{CF^3}{CE^2} \cdot \frac{ER}{CR} \]

But because \( CF \) is \( \frac{1}{2} \) of \( CE \), \( \frac{CF^3}{CE^2} \) is \( \frac{1}{2} \) \( CF^3 \)

\[ = \frac{CF^3}{2}, \text{ a constant quantity. Therefore the density of the light at } I \text{ is proportional to } \frac{ER}{CR} \text{ or to } \frac{AI}{bI}, \text{ because the points } R \text{ and } I \text{ are similarly situated in } EC \text{ and } AB. \]

Farther, if the semi-aperture \( CE \) of the lens be called \( l \),

\[ \frac{CF^3}{2} = \frac{1}{l}, \text{ and the density at } I \text{ is } \frac{AI}{8l}. \]

Here it is proper to observe, that since the point \( R \) has the same situation in the diameter \( EB \) that the point \( I \) has in the diameter \( AD \) of the circle of dispersion, the circle described on \( EB \) may be conceived as the magnified representation of the circle of dispersion. The point \( F \), for instance, represents the point \( f \) in the circle of dispersion, which bisects the radius \( bA \); and \( f \) receives no light from any part of the lens which is nearer the centre than \( F \), being illuminated only by the light which comes through \( EF \), and its opposite \( BF' \). The same may be said of every other point.

In like manner, the density of the light in \( f \), the middle between \( b \) and \( A \), is measured by \( \frac{EF}{CF} \), which is \( \frac{EF}{CF} \), or \( l \). This makes the density at this point a proper standard of comparison. The density there is to the density at \( I \) as \( \frac{AI}{bI} \), or as \( bI \) to \( AI \); and this is the simplest mode of comparison. The density half way from the centre of the circle of dispersion, is to the density at any point \( I \) as \( bI \) to \( AI \).

Lastly, through \( L \) describe the common rectangular hyperbola \( kLm \), meeting the ordinates of the former in \( k \), \( L \), and \( m \); and draw \( kb \) parallel to \( EC \), cutting the ordinates in \( g \), \( f \), \( r \), &c. Then \( CR : CE = Ek : RN \), and \( CR : CE = CR : Ek : RN - Ek \), or \( CR : RE = Ek : rn \), and \( bI : IA = Ek : rn \). And thus we have a very simple expression of the density in any point of the circle of dispersion. Let the point be anywhere, as at \( I \). Divide the lens in \( R \) as \( AD \) is divided in \( l \), and then \( rn \) is as the density in \( I \).

These two measures were given by Newton; the first in his treatise de Mundi Systemate, and the last in his Optics, but both without demonstration.

If the hyperbola \( kLm \) be made to revolve round the axis \( CQ \), it will generate a solid spindle, which will measure the whole quantity of light which passes through different portions of the circle of dispersion. Thus the solid produced by the revolution of \( Lk \) will measure all the light which occupies the outer part of the circle of dispersion lying without the middle of the radius. This space is \( \frac{1}{4} \)th of the whole circle; but the quantity of light is but \( \frac{1}{4} \)th of the whole.

A fully more simple expression of the whole quantity of light passing through different portions of the circle of chromatic dispersion may now be obtained as follows:

It has been demonstrated, that the density of the light at \( I \) is as \( \frac{AI}{bI} \), or as \( \frac{ER}{CR} \). Suppose the figure to turn round the axis: the points \( I \) and \( R \) describe circumferences of circles; and the whole light passing through this circumference is as the circumference, or as the radius and as the density... It is therefore as \( \frac{ER}{CR} \), that is, as \( ER \). Draw any straight line \( EM \), cutting \( RN \) in \( s \), and any other ordinate \( FL \) in \( xRs \). The whole light which illuminates the circumference described by \( T \) is to the whole light which illuminates the centre \( b \) as \( ER \) to \( EC \), or as \( Rs \) to \( CM \). In like manner, the whole light which illuminates the circumference described by the point \( f \) in the circle of dispersion, is to the whole light which illuminates the centre \( b \) as \( Fx \) to \( CM \). The lines \( CM, Rs, Fx \), are therefore proportional to the whole light which illuminates the corresponding circumferences in the circle of dispersion. Therefore the whole light which falls on the circle whose radius is \( bI \), will be represented by the trapezium in \( CRs \); and the whole light which falls on the ring described by \( IA \), will be represented by the triangle \( EsR \); and so of any other portions.

By considering the figure, we see that the distribution of the light is exceedingly unequal. Round the margin it has no sensible density; while its density in the very centre is incomparably greater than in any other point, being expressed by the asymptote of a hyperbola. Also the circle described with the radius \( \frac{Ab}{2} \) contains this of the whole light. No wonder, then, that the confusion caused by the mixture of these circles of dispersion is less than one should expect; besides, it is evident that the most lively or impressive colours occupy the middle of the spectrum, and are there much denser than the rest. The margin is covered with an illumination of deep red and violet, neither of which colours is brilliant. The margin will be of a dark claret colour. The centre revives all the colours, but in a proportion of intensity greatly different from that in the common prismatic spectrum, because the radiant points \( L, p, b, g, \) etc., by which it is illuminated, are at such different distances from it. It will be white; but, we apprehend, not a pure white, being greatly overcharged with the middle colours.

These considerations show that the coloured fringes which are observed to border very luminous objects seen on a dark ground through optical instruments, do not proceed from the object-glass of a telescope or microscope, but from an improper construction of the eye-glasses. The chromatic dispersions would produce fringes of a different colour when they produce any at all, and the colours would be differently disposed. But this dispersion by the object-glass can hardly produce any fringes: its effect is a general and almost uniform mixture of circles all over the field, which produces an uniform haziness, as if the object were viewed at an improper distance, or out of its focus, as we vulgarly express it.

We may at present form a good guess at the limit which this cause puts to the performance of a telescope. A point of a very distant object is represented, in the picture formed by the object-glass, by a little circle, whose diameter is at least \( \frac{1}{300} \)th of the aperture of the object-glass, making a very full allowance for the superior brilliancy and density of the central light. We look at this picture with a magnifying eye-glass. This magnifies the picture of the point. If it amplifies it to such a degree as to make it an object individually distinguishable, the confusion is then sensible. Now this can be computed. An object subtending one minute of a degree is distinguished by the dullest eye, even although it be a dark object on a bright ground. Let us therefore suppose a telescope, the object-glass of which is of six feet focal distance, and one inch aperture. The diameter of the circle of chromatic dispersion will be \( \frac{1}{300} \)th of an inch, which subtends at the centre of the object-glass an angle of about nine seconds and a half. This, when magnified six times by an eye-glass, would become a dis-

tinguishable object; and a telescope of this length would be indistinct if it magnified more than six times, if a point were thus spread out into a spot of uniform intensity. But the spot is much less intense about its margin. It is found experimentally that a piece of engraving, having fine cross hatches, is not sensibly indistinct till brought so far from the limits of perfectly distinct vision, that this indistinctness amounts to \( 6' \) or \( 5' \) in breadth. Therefore such a telescope will be sensibly distinct when it magnifies thirty-six times; and this is very agreeable to experience.

We come, in the second place, to the more arduous task of ascertaining the error arising from the spherical figure of the surfaces employed in optical instruments. Suffice it to say, before we begin, that although geometers have exhibited other forms of lenses which are totally exempt from this error, they cannot be executed by the artist; and we are therefore restricted to the employment of spherical surfaces. We shall set out with Dr Smith's fundamental theorem.

### 1. In Reflections.

Let \( AVB \) (fig 3) be a concave spherical mirror, of which \( C \) is the centre, \( V \) the vertex, \( CV \) the axis, and \( F \) the focus of an infinitely slender pencil of parallel rays passing through the centre. Let the ray \( aA \), parallel to the axis, be reflected in \( AG \), crossing the central ray \( CV \) in \( f \). Let \( AP \) be the sine of the semi-aperture \( AV, AD \) its tangent, and \( CD \) its secant.

The aberration \( Ef \) from the principal focus of central rays is equal to \( \frac{1}{2} VD \), the excess of the secant above the radius, or very nearly equal to \( \frac{1}{2} \) of \( VP \), the versed sine of the semi-aperture.

For, because \( AD \) is perpendicular to \( CA \), the points \( C, A, D \), are in a circle, of which \( CD \) is the diameter; and because \( Af \) is equal to \( Cf \), by reason of the equality of the angles \( fAC, fCA, \) and \( CAa, f \) is the centre of the circle through \( C, A, D \), and \( fD \) is \( \frac{1}{2} CD \). But \( FC \) is \( \frac{1}{2} CV \). Therefore \( Ef \) is \( \frac{1}{2} VD \).

But because \( DV : VP = DC : VC \), and \( DC \) is very little greater than \( VC \) when the aperture \( AB \) is moderate, \( DV \) is very little greater than \( VP \), and \( Ef \) is very nearly equal to \( \frac{1}{2} VP \).

**Cor. 1.** The longitudinal aberration \( Ef \) is \( \frac{AV^2}{4CV} \), for \( PV \) is very nearly \( \frac{AV^2}{2CV} \).

**Cor. 2.** The lateral aberration \( FG \) is \( \frac{AV^2}{2CV} \). For \( FG : Ef = AP : Pf = AV : \frac{1}{2} CV \) nearly, and therefore \( FG = \frac{AV^2}{4CV} \cdot \frac{2}{CV} = \frac{AV^2}{2CV^2} \).

### 2. In Refractions.

Let \( AVB \) (fig. 4 or 5) be a spherical surface separating... two refracting substances, C the centre, V the vertex, AV the semi-aperture, AP its sine, PV its versed sine, and F the focus of parallel rays infinitely near to the axis. Let the extreme ray \(aA\), parallel to the axis, be refracted into AG, crossing CF in \(f\), which is therefore the focus of extreme parallel rays.

The rectangle of the sine of incidence, by the difference of the sines of incidence and refraction, is to the square of the sine of refraction, as the versed sine of the semi-aperture is to the longitudinal aberration of the extreme rays.

Call the sine of incidence \(i\), the sine of refraction \(r\), and their difference \(d\).

Join CA, and about the centre \(f\) describe the arch AD.

The angle ACV is equal to the angle of incidence, and CAM is the angle of refraction. Then, since the sine of incidence is to the sine of refraction as VF to CF, or as AF to CF; that is, as DF to CF, we have

\[ \frac{CF}{FV} = \frac{CF}{fD}, \]

by conversion

\[ \frac{CF}{CV} = \frac{CF}{CD}, \]

altern. conver. \(CF - CF : CV - CD = CF : CV \)

or

\[ \frac{EF}{VD} = \frac{CF}{CV} = r : d. \]

Now \(PV = \frac{AP^2}{CP + CV} = \frac{AP^2}{2CV}\) nearly, and \(PD = \frac{AP^2}{fP + fV} = \frac{AP^2}{2fV}\) nearly, \(= \frac{AP^2}{2FV}\) nearly. Therefore \(PV : PD = FV : CV\), and \(DV : PV = CF : FV\) nearly.

We had above \(Ff : VD = r : d\),

and now \(VD : PV = CF : FV = r : i\);

therefore \(Ff : PV = r^2 : di\),

and \(Ff = \frac{r^2}{di} \cdot PV.\) Q.E.D.

The aberration will be different according as the refraction is made towards or from the perpendicular; that is, according as \(r\) is less or greater than \(i\). They are in the ratio of \(r^2\) to \(di\), or of \(r^3\) to \(i^2\). The aberration therefore is always much diminished when the refraction is made from a rare into a dense medium. The proportion of the sines for air and glass is nearly that of 3 to 2. When the light is refracted into the glass, the aberration is nearly \(3/4\) of PV; and when the light passes out of glass into air, it is about \(3/8\) of PV.

Cor. 1. \(Ff = \frac{r^2}{di} \cdot \frac{AP^2}{2CV}\) nearly, and it is also \(= \frac{r^2}{di} \cdot \frac{AP^2}{2FV}\) because \(PV = \frac{AP^2}{2CV}\) nearly, and \(i : d = FV : CV\).

Cor. 2. Because \(fP : PA = Ff : FG\)

or \(FV : AV = Ff : FG\) nearly,

we have \(FG\), the lateral aberration, \(= Ff \times \frac{AV}{FV} = \frac{r^2}{di} \cdot \frac{AV^3}{2FV^2} = \frac{r^2}{i^2} \cdot \frac{AV^3}{2CV^2}\).

Cor. 3. Because the angle \(FAf\) is proportional to \(FG\)

very nearly, we have the angular aberration \(FAf = \frac{AV^3}{2FV^2} = \frac{r^2}{i^2} \cdot \frac{AV^3}{2CV^2}\).

In general, the longitudinal aberrations from the focus of telescope central parallel rays are as the squares of the apertures directly, and as the focal distances inversely; and the lateral aberrations are as the cubes of the apertures directly, and the squares of the focal distances inversely; and the angular aberrations are as the cubes of the aperture directly, and the cubes of the focal distances inversely.

The reader must have observed, that to simplify the investigation, some small errors are admitted. PV and PD are not in the exact proportion that we assumed them, nor is DF equal to FV. But in the small apertures which suffice for optical instruments, these errors may be disregarded.

This spherical aberration produces an indistinctness of vision, in the same manner as the chromatic aberration does, viz. by spreading out every mathematical point of the object into a little spot in its picture; which spots, by mixing with each other, confuse the whole. We must now determine the diameter of the circle of diffusion, as we did in the case of chromatic dispersion.

Let a ray \(Ba\) (fig. 6) be refracted on the other side of the axis, into \(eHf\), cutting \(aG\) in \(H\), and draw the perpendicular EH. Put \(AV = a, AV = a, VF\) (or \(VF\) or \(Vg\), which in this comparison may be taken as equal) \(= f\),

\(Ff = b,\) and \(FE = x.\)

\(AV^2 : aV^2 = Ff : Ff\) (already demonstrated), and \(Ff = \frac{a^2}{b^2}b,\) and \(Ff - Ff\) (or \(f^2\)) \(= b - \frac{a^2}{b^2}b = \frac{a^2b - a^2b}{a^2} = \frac{b}{a^2}\)

\((a^2 - a^2) = \frac{b}{a}(a + a)(a - a).\) Also \(Ff : PA = Ff : EH,\)

or \(f : a = x : \frac{ax}{f} = EH.\) And \(Pr : Pp = EH : Ep\) or \(a : f = \frac{ax}{f} : \frac{ax}{a} = Ep.\) Therefore \(f^2 = \frac{ax}{a} + x = \frac{ax + ax}{a} = \frac{x}{a}(a + a).\) Therefore \(x : a(a + a) = \frac{b}{a^2}(a + a)(a - a),\)

and \(x = \frac{b}{a^2}(a - a),\) and \(x = \frac{b}{a^2}(a - a).\) Therefore \(x\) is greatest when \(a(a - a)\) is greatest; that is, when \(a = \frac{1}{2}a.\)

Therefore EH is greatest when \(Pr\) is equal to the half of AP. When this is the case, we have at the same time \(b : a(a - a) = \frac{b}{a^2} : \frac{1}{2}a^2,\) and \(x = \frac{1}{2}b,\) or \(EH = \frac{1}{2}FG.\)

That is, the diameter of the circle of aberration through which the whole of the refracted light must pass, is \(\frac{1}{2}\) of the diameter of the circle of aberration at the focus of parallel central rays. In the chromatic aberration there was \(\frac{1}{2}\); so that in this respect the spherical aberration does not create so great confusion as the chromatic.

We are now able to compare them, since we have the measure of both the circles of aberration.

It has not been found possible to give more than four inches of aperture to an object-glass of 100 feet focal distance, so as to preserve sufficient distinctness. If we compute the diameter of the circle EH corresponding to this aperture, we shall find it not much to exceed $\frac{1}{120,000}$ of an inch. If we restrict the circle of chromatic dispersion to $\frac{1}{36}$ of the aperture, which is hardly the fifth part of the whole dispersion in it, it is $\frac{1}{624}$ of an inch, and is about 1900 times greater than the other.

The circle of spherical aberration of a plano-convex lens, with the plane side next the distant object, is equal to the circle of chromatic dispersion when the semi-aperture is about 15°; for we saw formerly that EH is $\frac{1}{2}$ of FG, and that FG is $\frac{AP^2}{AC^2}$, and therefore EG = $\frac{AP^2}{AC^2}$. This being made $= \frac{AP}{AC}$ gives us $AP = \sqrt{\frac{55}{55}}$, which is nearly $\frac{AC}{4}$, and corresponds to an aperture of 30° diameter, if r be to i as 3 to 2.

Sir Isaac Newton was therefore well entitled to say, that it was quite needless to attempt figures which should have less aberration than spherical ones, while the confusion produced by the chromatic dispersion remained uncorrected. Since the indistinctness is as the squares of the diameters of the circles of aberration, the disproportion is quite beyond our imagination, even when Newton has made such a liberal allowance to the chromatic dispersion. But it must be acknowledged, that he has not attended to the distribution of the light in the circle of spherical aberration, and has hastily supposed it to be like the distribution of the coloured light, indefinitely rare in the margin, and denser in the centre.

Boscovich has shown that the light in the margin of the circle of spherical aberration, instead of being incomparably rarer than in the spaces between it and the centre, is incomparably denser. The indistinctness therefore produced by the intersection of these luminous circumferences is vastly great, and increases the whole indistinctness exceedingly. By a gross calculation which we made, it appears to be increased at least 500 times. The proportional indistinctness, therefore, instead of being 1900$^2$ to 1, is only 1900$^2$ to 500, or nearly 7220 to 1; a proportion still sufficiently great to warrant Newton's preference of the reflecting telescope of his invention. And we may now observe, that the reflecting telescope has even a great advantage over a refracting one of the same focal distance, with respect to its spherical aberration: For we have seen (Cor. 2) that the lateral aberration is $\frac{AV^2}{CV^2}$. This for a plano-convex glass is nearly $\frac{9}{4} \cdot \frac{AV^2}{CV^2}$, and the diameter of the circle of aberration is one fourth of this, or $\frac{9}{16} \cdot \frac{AV^2}{CV^2}$. In like manner, the lateral aberration of a concave mirror is $\frac{AV^2}{CV^2}$; and the diameter of the circle of dispersion is $\frac{AV^2}{CV^2}$; and therefore, if the surfaces were portions of the same sphere, the diameter of the circle of aberration of refracted rays would be to that of the circle of aberration of reflected rays as $\frac{9}{16}$ to $\frac{1}{4}$, or as 9 to 4. But when the refracting and reflecting surfaces, in the position here considered, have the same focal distance, the radius of the refracting surface is four times that of the reflecting surface. The proportion of the diameters of the circles of spherical aberration is that of $9 \times 4^2$ to 4, or of 144 to 4, or 36 to 1. The distinctness therefore of the reflector is 36 × 36, or 1296 times greater than that of a plano-convex lens (placed with the plane side next the distant object) of the same breadth and focal distance, and will therefore admit of a much greater magnifying power. This comparison is indeed made in circumstances most favourable to the reflector, because this is the very worst position of a plano-convex lens. But we have not as yet learned the aberration in any other position. In another position the refraction and consequent aberration of both surfaces are complicated.

Before we proceed to the consideration of this very difficult subject, we may deduce, from what has been already demonstrated, several general rules and maxims in the construction of telescopes, which will explain (to such readers as do not wish to enter more deeply into the subject) and justify the proportion which long practice of the best artists has sanctioned.

Indistinctness proceeds from the commixture of the circles of aberration on the retina of the eye: for any one sensible point of the retina, being the centre of a circle of aberration, will at once be affected by the admixture of the rays of as many different pencils of light as there are sensible points in the area of that circle, and will convey to the mind a mixed sensation of as many visible points of the object. This number will be as the area of the circle of aberrations, whatever be the size of a sensible point of the retina. Now in vision with telescopes the diameter of the circle of aberration on the retina is as the apparent magnitude of the diameter of the corresponding circle in the focus of the eye-glass; that is, as the angle subtended by this diameter at the centre of the eye-glass; that is, as the diameter itself directly, and as the focal distance of the eye-glass inversely. And the area of that circle on the retina is as the area of the circle on the focus of the eye-glass directly, and as the square of the focal distance of the eye-glass inversely. And this is the measure of the apparent indistinctness.

Cor. In all sorts of telescopes, and also in compound microscopes, an object is seen equally distinct when the focal distances of the eye-glasses are proportional to the diameters of the circles of aberration in the focus of the object-glass.

Here we do not consider the trifling alteration which well-constructed eye-glasses may add to the indistinctness of the first image.

In refracting telescopes, the apparent indistinctness is as the area of the object-glass directly, and as the square of the focal distance of the eye-glass inversely. For it has been shown, that the area of the circle of dispersion is as the area of the object-glass, and that the spherical aberration is insignificant when compared with this.

Therefore, to make reflecting telescopes equally distinct, the diameter of the object-glass must be proportional to the focal distance of the eye-glass.

But in reflecting telescopes, the indistinctness is as the sixth power of the aperture of the object-glass directly, and as the fourth power of the focal distance of the object-glass and square of the focal distance of the eye-glass inversely. This is evident from the dimensions of the circle of aberration, which was found proportional to $\frac{AV^2}{CV^2}$.

Therefore, to have them equally distinct, the cubes of the apertures must be proportional to the squares of the focal distance multiplied by the focal distance of the eye-glass.

By these rules, and a standard telescope of approved goodness, an artist can always proportion the parts of any instrument he wishes to construct. Huyghens made one, of which the object-glass had 80 feet focal distance, and three inches diameter; the eye-glass had 33 inches focal distance; and its performance was found superior to any which he had seen. Nor did this appear owing to any chance goodness of the object-glass, because he found others equally good which were constructed on similar proportions. This has therefore been adopted as a standard.

It does not at first appear how there can be any difficulty in this matter, because we can always diminish the aperture of the object-glass or speculum, till the circle of aberration is as small as we please. But by diminishing this aperture, we diminish the light in the duplicate ratio of the aperture. Whatever be the aperture, the brightness is diminished by the magnifying power, which spreads the light over a greater surface in the bottom of the eye. The apparent brightness must be as the square of the aperture of the telescope directly, and the square of the amplification of the diameter of an object inversely. Objects therefore will be seen equally bright if the apertures of the telescopes be as the focal distances of the object-glasses directly, and the focal distances of the single eye-glass (or eye-glass equivalent to the eye-piece) inversely. Therefore, to have telescopes equally distinct and equally bright, we must combine these proportions with the former. It is needless to go farther into this subject, because the construction of refracting telescopes has been so materially changed by the correction of the chromatic aberration, that there can hardly be given any proportion between the object-glass and eye-glasses. Every thing now depends on the degree in which we can correct the aberrations of the object-glass. We have been able so far to diminish the chromatic aberration, that we can give very great apertures without its becoming sensible. But this is attended with so great an increase of the aberration of figure, that this last becomes a sensible quality. A lens which has 30° for its semi-aperture, has a circle of aberration equal to its chromatic aberration. Fortunately we can derive from the very method of contrary refractions, which we employ for removing the chromatic aberrations, a correction of the other. For this contrivance we are also indebted to the illustrious Newton.

We call this Newton's contrivance, because he was the first who proposed a construction of an object-glass in which the aberration was corrected by the contrary aberrations of glass and water.

Huyghens had indeed supposed, that our all-wise Creator had employed in the eyes of animals many refractions in place of one, in order to make the vision more distinct; and the invidious detractors from Newton's fame have catched at this vague conjecture as an indication of his knowledge of the possibility of destroying the aberration of figure by contrary refractions. But this is very ill founded. Huyghens has acquired sufficient reputation by his theory of aberrations. The scope of his writing in the passage alluded to, is to show that, by dividing any intended refraction into parts, and producing a certain convergence to or divergence from the axis of an optical instrument, by means of two or three lenses instead of one, we diminish the aberrations four or nine times. This conjecture about the eye was therefore in the natural train of his thoughts. But he did not think of destroying the aberration altogether by opposite refractions. Newton, in 1669, says that opticians need not trouble themselves about giving figures to their glasses other than spherical. If this figure were all the obstacle to the improvement of telescopes, he could show them a construction of an object-glass having spherical surfaces where the aberration is destroyed; and accordingly he gives the construction of one composed of glass and water, in which this is done completely by means of contrary refractions.

The general principle is this. When the radiant point R (fig. 7), or focus of incident rays, and its conjugate focus F of refracted central rays, are on opposite sides of the refracting surface or lens V, the conjugate focus f of marginal rays is nearer to R than F is. But when the focus of incident rays R' lies on the same side with its conjugate focus F' for central rays, R'f is greater than R'F'.

Now fig. 8 represents the contrivance for destroying the colour produced at F', the principal focus of the convex lens V, of crown glass, by means of the contrary refraction of the concave lens v of flint glass. The incident parallel rays are made to converge to F by the first lens. This convergence is diminished, but not entirely destroyed, by the concave lens v, and the focus is formed in F'. F and F' therefore are conjugate foci of the concave lens. If F be the focus of V for central rays, the marginal rays will be collected at some point f nearer to the lens. If F' be now considered as the focus of light incident on the centre of v, and F' be the conjugate focus, the marginal ray pF' would be refracted to some point f' lying beyond F'. Therefore the marginal ray pF may be refracted to F, if the aberration of the concave be properly adjusted to that of the convex.

This brings us to the most difficult part of our subject, the compounded aberrations of different surfaces. Our limits will not admit of our treating this in the same elementary and perspicuous manner that we employed for a single surface. We must try to do it in a compendious way, which will admit at once the different surfaces, and the different refractive powers of different substances. This must naturally render the process more complicated; but we hope to treat the subject in a way easily comprehended by any person moderately acquainted with common algebra.

Lemma 1. In the right-angled triangle MXS (fig. 9), of which one side MX is very small in comparison of either of the others, the excess of the hypotenuse MS above the side XS is very nearly equal to \(\frac{MX^2}{2MS}\) or \(\frac{MX^2}{2XS}\). For if about the centre S, with the radius SM, we describe the semicircle AMO, we have AX·XO = MX². Now AX = MS - SX, and XO is nearly equal to 2MS or 2XS; on the other hand, MS is nearly equal to XS + \(\frac{MX^2}{2XS}\); and in like manner MG is nearly equal to \(\frac{MX}{2XG} + XG\), and MH is nearly equal to \(\frac{MX^2}{2XH} + XH\).

Prop. I. Let the ray mM, incident on the spherical surface AM, converge to G; that is, let G be the focus of incident rays. It is required to find the focus H of refracted rays.

Let \(m\) express the ratio of the sine of incidence and refraction; that is, let \(m\) be to 1 as the sine of incidence to the sine of refraction in the substance of the sphere. MG : GS = sin. MSH : sin. SMG, and \( m : 1 = \sin. SMG : \sin. SMH; \) therefore \( m \cdot MG : GS = \sin. MSH : \sin. SMH. \)

Now S, MSH : S, SMH = MH : HS. Therefore, finally, \( m \cdot MG : GS = MH : HS. \)

Now, let MS, the radius of the refracting surface, be called \( a \). Let AG, the distance of the focus of incident rays from the surface, be called \( r \). And let AH, the focal distance of refracted rays, be called \( x \). Lastly, let the sine MX of the semi-aperture be called \( e \). Observe, too, that \( a, r, x \) are to be considered as positive quantities, when AS, AG, AH lie from the surface in the direction in which the light is supposed to move. If therefore the refracting surface be concave, that is, having the centre on that side from which the light comes; or if the incident rays are divergent, or the refracted rays are divergent; then \( a, r, x \) are negative quantities.

It is plain that \( HS = x - a; GS = r - a; \) also \( \Lambda X = \frac{e^2}{2a} \) nearly. \( HX = x - \frac{e^2}{2a}; GX = r - \frac{e^2}{2a}. \) Now add to \( HX \) and to \( GH \) their differences from \( MH \) and \( MG, \) which (by the Lemma) are \( \frac{e^2}{2x} \) and \( \frac{e^2}{2r}. \) We get \( MH = x - \frac{e^2}{2a} + \frac{e^2}{2x}, \) and \( MG = r - \frac{e^2}{2a} + \frac{e^2}{2r}. \) In order to shorten our notation, make \( k = \frac{1}{a} - \frac{1}{r}. \) This will make \( MG = r - \frac{ke^2}{2a}. \)

Now substitute these values in the analogy \( MH : HS = m \cdot MG : GS; \) it becomes \( x - \frac{e^2}{2a} + \frac{e^2}{2x} : x - a = mr - \frac{mke^2}{2a} : r - a \) (or \( ark), \) because \( k = \frac{r - a}{ark}, \) and \( ark = r - a. \) Now multiply the extreme and mean terms of this analogy. It is evident that it must give us an equation which will give us a value of \( x \) or \( AH, \) the quantity sought.

But this equation is quadratic. We may avoid the solution by an approximation which is sufficiently accurate, by substituting for \( x \) in the fraction \( \frac{e^2}{2x} \) (which is very small in all cases of optical instruments) an approximate value very easily obtained, and very near the truth. This is the focal distance of an infinitely slender pencil of rays converging to G. This we know by the common optical theorem to be \( \frac{amr}{(m-1)r-a}. \) Let this be called \( \varphi; \) if we substitute \( k \) in place of \( \frac{1}{a} - \frac{1}{r}, \) this value of \( \varphi \) becomes \( \frac{am}{m-ak}. \)

This gives us, by the by, an easily-remembered expression (and beautifully simple) of the refracted focus of an infinitely slender pencil, corresponding to any distance \( r \) of the radiant point. For since \( \varphi = \frac{am}{m-ak}; \) \( \varphi \) must be \( \frac{m-ak}{am} = \frac{m-ak}{am} = \frac{1}{a} - \frac{k}{m}. \) We may even express it more simply, by expanding \( k, \) and it becomes \( \varphi = \frac{1}{a} - \frac{1}{ma} - \frac{1}{mr}. \)

Now put this value of \( \frac{1}{\varphi} \) in place of \( \frac{1}{x} \) in the analogy employed above. The first term of the analogy becomes

\[ x - \frac{e^2}{2a} + \frac{e^2}{2a} - \frac{ke^2}{2a}, \text{ or } x - \frac{ke^2}{2a}. \]

The analogy now becomes \( x - \frac{ke^2}{2a} : x - a = mr - \frac{mke^2}{2a} : ark. \) Hence we obtain the linear equation \( mrx - \frac{mke^2x}{2} = mra + \frac{mke^2}{2} = arkx - \frac{ark^2e^2}{2a}; \) from which we finally deduce

\[ x = \frac{mra - \frac{1}{2}make^2 - \frac{ark^2e^2}{2a}}{mr - ark - \frac{1}{2}mke^2}. \]

We may simplify this greatly by attending to the elementary theorem in fluxions, that the fraction \( \frac{x + dx}{y + dy} \) differs from the fraction \( \frac{x}{y} \) by the quantity \( \frac{ydx + xdy}{y^2}; \) this being the fluxion of \( \frac{x}{y}. \) Therefore \( \frac{x + dx}{y + dy} = \frac{x}{y} + \frac{ydx + xdy}{y^2}. \)

Now the preceding formula is nearly in this situation. It may be written thus:

\[ \frac{mra - \frac{1}{2}make^2 + \frac{ark^2e^2}{2a}}{mr - ark - \frac{1}{2}mke^2}. \]

Here the last terms of the numerator and denominator are very small in comparison with the first, and may be considered as the \( dx \) and \( dy, \) while \( mra \) is the \( x, \) and \( mr - ark \) is the \( y. \) Treating it in this way, it may be stated thus:

\[ x = \frac{ma}{m-ak} + \frac{(mra)mk - (mr-ark)(mka + arke^2)}{r^2(m-ak)^3}. \]

The first term \( \frac{ma}{m-ak} \) is evidently \( \varphi, \) the focal distance of an infinitely slender pencil. Therefore the aberration is expressed by the second term, which we must endeavour to simplify.

Now the numerator of the small fraction, by multiplication, may be expressed thus:

\[ m^2a^2 \left( \frac{rk^2}{m} - \frac{rk^2}{m^2a} + \frac{rk^2}{m^3} \right) \frac{1}{2}e^2. \]

The denominator is \( r^2(m-ak)^3, \) and the fraction now becomes \( \frac{m^2a^2}{r^2(m-ak)^3} \left( \frac{rk^2}{m} - \frac{rk^2}{m^2a} + \frac{rk^2}{m^3} \right) \frac{1}{2}e^2, \) which is evidently \( \varphi^2 \left( \frac{rk^2}{m} - \frac{rk^2}{m^2a} + \frac{rk^2}{m^3} \right) \frac{1}{2}e^2. \)

Now recollect that \( k = \frac{1}{a} - \frac{1}{r}. \) Therefore \( k^3 = \frac{k^3}{m} \left( \frac{1}{a} - \frac{1}{r} \right) = \frac{k^3}{m} - \frac{k^3}{m^2r}. \) Therefore, instead of \( \frac{k^3}{m^2a}, \) write \( \frac{k^3}{m^2} - \frac{k^3}{m^2r}, \) and we get the fraction \( \varphi^2 \left( \frac{k^3}{m^2} - \frac{k^3}{m^2r} + \frac{k^3}{m^3} \right) \frac{1}{2}e^2 = \varphi^2 \left( \frac{k^3}{m^2} - \frac{mk^3}{m^2r} + \frac{mk^3}{m^3} \right) \frac{1}{2}e^2, \) which is equal to \( \varphi^2 \left( \frac{1}{m} \left( \frac{k^3}{m} - \frac{mk^3}{r} \right) \right) \frac{1}{2}e^2, \) and finally to \( \varphi^2 \left( \frac{1}{m} \left( \frac{k^3}{m} - \frac{mk^3}{r} \right) \right) \frac{1}{2}e^2. \)

Therefore the focal distance of refracted rays is

\[ x = \varphi - \varphi^2 \left( \frac{1}{m} \left( \frac{k^3}{m} - \frac{mk^3}{r} \right) \right) \frac{1}{2}e^2. \]

This consists of two parts. The first, \( \varphi, \) is the focal dis- Telescope. tance of an infinitely slender pencil of central rays, and the other, \( \frac{m}{m^2} \left( k^2 - \frac{mk^2}{r} \right) e^2 \) is the aberration arising from the spherical figure of the refracting surface.

Thus our formula has at last assumed a very simple form, and is vastly preferable to Dr Smith's for practice.

This aberration is evidently proportional to the square of the semi-aperture, and to the square of the distance \( \varphi \); but in order to obtain this simplicity, several quantities were neglected. The assumption of the equality of \( AX \) to \( \frac{e^2}{2a} \) is the first source of error. A much more accurate value of it would have been \( \frac{2ae^2}{4a^2 + e^2} \), for it is really

\[ = \frac{e^2}{2a - AX}. \]

If for \( AX \) we substitute its approximated value \( \frac{e^2}{2a} \), we should have \( AX = \frac{e^2}{2a} = \frac{2ae^2}{4a^2 - e^2} \). To have used this value would not much have complicated the calculus; but it did not occur to us till we had finished the investigation, and it would have required the whole to be changed. The operation in page 149, col. 2, par. 1, is another source of error. But these errors are very inconsiderable when the aperture is moderate. They increase for the most part with an increase of aperture, but not in the proportion of any regular function of it; so that we cannot improve the formula by any manageable process, and must be contented with it. The errors are precisely the same with those of Dr Smith's theorem, and indeed with those of any that we have seen, which are not vastly more complicated.

As this is to be frequently combined with subsequent operations, we shorten the expression by putting \( \varphi \) for \( \frac{m}{m^2} \left( k^2 - \frac{mk^2}{r} \right) e^2 \). Then \( \varphi \) will express the aberration of the first refraction from the focal distance of an infinitely slender pencil; and now the focal distance of refracted rays is \( f = \varphi - \varphi' \).

If the incident rays are parallel, \( r \) becomes infinite, and \( \varphi = \frac{m-1}{m} \frac{k^2}{2} \). But in this case \( k \) becomes \( \frac{1}{a} \) and \( \frac{1}{\varphi} = \frac{ma}{m-1} \), and \( \varphi' \) becomes \( \frac{m^2 a^2}{(m-1)^2} \times \frac{m-1}{m^2} \times \frac{1}{a^2} \times \frac{e^2}{2} = \frac{e^2}{(2m-1)ma} \). This is the aberration of extreme parallel rays.

We must now add the refraction of another surface.

Lemma 2. If the focal distance \( AG \) be changed by a small quantity \( Gg \), the focal distance \( AH \) will also be changed by a small quantity \( Hh \), and we shall have

\[ m \cdot AG^2 : AH^2 = Gg : Hh. \]

Draw \( Mg, Mh \), and the perpendiculars \( Gi, Hk \). Then, because the sines of the angles of incidence are in a constant ratio to the sines of the angles of refraction, and the increments of these small angles are proportional to the increments of the sines, these increments of the angles are in the same constant ratio. Therefore we have the angle \( GMg \) to \( HMh \) as \( m \) to \( 1 \).

Now \( Gg : Gi = AG : AM \),

and \( Gi : Hk = m \cdot AG : HA \),

and \( Hk : Hh = MA : AH \);

therefore \( Gg : Hh = m \cdot AG^2 : AH^2 \).

The easiest and most perspicuous method for obtaining the aberration of rays twice refracted, will be to consider the first refraction as not having any aberration, and determine the aberration of the second refraction. Then con-

ceive the focus of the first refraction as shifted by the aberration. This will produce a change in the focal distance of the second refraction, which may be determined by this Lemma.

Prop. II. Let \( AM, BN \) (fig. 10) be two spherical surfaces, including a refracting substance, and having their centres \( C \) and \( c \) in the line \( AG \). Let the ray \( aA \) pass through the centres, which it will do without refraction. Let another ray \( mM \), tending to \( G \), be refracted by the first surface into \( MH \), cutting the second surface in \( N \); where it is farther refracted into \( NI \). It is required to determine the focal distance \( BI \).

It is plain that the sine of incidence on the second surface is to the sine of refraction into the surrounding air as \( 1 \) to \( m \). Also \( BI \) may be determined in relation to \( BH \), by means of \( BH, Nz, Be \), and \( \frac{1}{m} \), in the same way that \( AH \) was determined in relation to \( AG \), by means of \( AG, MX, AC \), and \( m \).

Let the radius of the second surface be \( b \), and let \( e \) still express the semi-aperture (because it hardly differs from \( NZ \)). Also let \( a \) be the thickness of the lens. Then observe, that the focal distance of the rays refracted by the first surface (neglecting the thickness of the lens and the aberration of the first surface), is the distance of the radiant point for the second refraction, or is the focal distance of rays incident on the second surface. In place of \( r \) therefore we must take \( \varphi \); and as we made \( k = \frac{1}{a} - \frac{1}{r} \) in order to abbreviate the calculus, let us now make \( l = \frac{1}{b} - \frac{1}{\varphi} \); and make \( \frac{1}{f} = \frac{1}{b} - ml \), as we made \( \frac{1}{\varphi} = \frac{1}{a} - \frac{1}{m} \). Lastly, in place of \( \varphi = \frac{m-1}{m} \frac{k^2}{2} \), make \( \varphi' = \left( \frac{1}{m} - 1 \right) \frac{e^2}{2} \left( \frac{P}{m} - \frac{F}{m} \right) \frac{e^2}{2} = \frac{m-1}{m} \frac{m^2 F - m^2 E}{\varphi} \frac{e^2}{2} \).

Thus we have got an expression similar to the other; and the focal distance \( BI \), after two refractions, becomes \( BI = f - f' \).

But this is on the supposition that \( BH \) is equal to \( \varphi \), whereas it is really \( \varphi - \varphi' \). This must occasion a change in the value just now obtained of \( BI \). The source of the change is twofold. 1st, Because in the value \( \frac{1}{b} - \frac{1}{\varphi} \) we must put \( \frac{1}{b} - \frac{1}{\varphi - \varphi'} \), and because we must do the same in the fraction \( \frac{m^2 F}{\varphi} \). In the second place, when the value of \( BH \) is diminished by the quantity \( \varphi' + a \), \( BI \) will suffer a change in the proportion determined by the second Lemma. The first difference may safely be neglected, because the value of \( \varphi' \) is very small, by reason of the coefficient \( \frac{e^2}{2} \) being very small, and also because the variation bears a very small ratio to the quantity itself, when the true value of \( \varphi \) differs but little from that of the quantity for which it is employed. The chief change in \( BI \) is that which is determined by the Lemma. Therefore take from \( BI \) the variation of BH, multiplied by \( \frac{mBP}{BH} \), which is very nearly

\[ = \frac{mf^2}{\varphi^2}. \]

The product of this multiplication is \( mf^2 \theta + \frac{mf^2 \alpha}{\varphi^2} \).

This being taken from \( f \), leaves us for the value of BI,

\[ f = \frac{f^2 ma}{\varphi^2} - f^2 (m \theta + \theta'). \]

In this value \( f \) is the focal distance of an infinitely slender pencil of rays twice refracted by a lens having no thickness, \( \frac{mf^2}{\varphi^2} \) is the shortening occasioned by the thickness, and \( f^2 (m \theta + \theta') \) is the effect of the two aberrations arising from the aperture.

It will be convenient, for several collateral purposes, to exterminate from these formulae the quantities \( k, l, \) and \( \varphi \).

For this purpose, make \( \frac{1}{n} = \frac{1}{a} - \frac{1}{b} \). We have already

\[ k = \frac{1}{a} - \frac{1}{r}, \quad \text{and} \quad \varphi = \frac{1}{a} - \frac{ma}{mr}; \quad \text{and} \quad l = \frac{1}{b} - \frac{1}{\varphi} = \frac{1}{b} - \frac{1}{a} + \frac{1}{ma} - \frac{1}{mr}. \]

Now for \( \frac{1}{b} - \frac{1}{a} \) write \( -\frac{1}{n} \), and we get

\[ l = \frac{1}{ma} - \frac{1}{mr} - \frac{1}{n}. \]

Therefore \( \frac{1}{f} = \frac{1}{b} - ml \) (by construction, page 150, Prop. II.) becomes \( \frac{1}{b} - \frac{1}{a} + \frac{1}{r} + \frac{m}{n} \)

\[ = \frac{m}{n} + \frac{1}{r} - \frac{1}{n} = \frac{m-1}{n} + \frac{1}{r}. \]

This last value of \( \frac{1}{f} \) (the reciprocal of the focus of a slender pencil twice refracted), viz. \( \frac{m-1}{n} + \frac{1}{r} \), is the simplest that can be imagined, and makes \( n \) as a substitute for \( \frac{1}{a} - \frac{1}{b} \); a most useful symbol, as we shall frequently find in the sequel. It also gives a very simple expression of the focal distance of parallel rays, which we may call the principal focal distance of the lens, and distinguish it in future by the symbol \( p \); for the expression \( \frac{1}{f} = \frac{m-1}{n} + \frac{1}{r} \) becomes \( \frac{1}{p} = \frac{m-1}{n} \) when the incident light is parallel. And this gives us another very simple and useful measure of \( f \);

for \( \frac{1}{f} \) becomes \( \frac{1}{p} + \frac{1}{r} \). These equations, \( \frac{1}{f} = \frac{m-1}{n} + \frac{1}{r} \),

\( \frac{1}{p} = \frac{m-1}{n} \), and \( \frac{1}{f} = \frac{1}{p} + \frac{1}{r} \), deserve therefore to be made very familiar to the mind.

We may also take notice of another property of \( n \). It is half the radius of an isosceles lens, which is equivalent to the lens whose radii are \( a \) and \( b \); for suppose the lens to be isosceles, that is, \( a = b \); then \( n = \frac{1}{a} - \frac{1}{a} \). Now the second \( a \) is negative if the first be positive, or positive if the first be negative. Therefore \( \frac{1}{a} - \frac{1}{b} = \frac{a+b}{a^2} = \frac{a+a}{a^2} = \frac{2}{a} \) and \( \frac{1}{n} = \frac{2}{a} \), and \( n = \frac{a}{2} \). Now the focal distance of this lens is \( \frac{m-1}{n} \), and so is that of the other, and they are equivalent.

But, to proceed with our investigation, recollect that we had \( \delta = \frac{m-1}{m} \left( \frac{k^3 - k^2}{r} \right) \frac{e^2}{2} \). Therefore \( m \delta = \frac{m-1}{m} \left( \frac{k^3 - k^2}{r} \right) \frac{e^2}{2} \).

And \( \delta' \) was \( \frac{m-1}{m} \left( \frac{-m^3 P + m^2 F}{\varphi^2} \right) \frac{e^2}{2} \).

Therefore \( f^2 (m \theta + \theta') \), the aberration (neglecting the thickness of the lens), is \( f^2 \left( \frac{m-1}{m} \left( \frac{k^3 - k^2}{r} - m^3 P + m^2 F \right) \frac{e^2}{2} \right) \).

If we now write for \( h, l, \) and \( \varphi \), their values as determined above, performing all the necessary multiplications, and arrange the terms in such a manner as to collect in one sum the co-efficients of \( a, n, \) and \( r \), we shall find four terms for the value of \( m \theta \), and ten for the value of \( \theta' \). The four are destroyed by as many with contrary signs in the value of \( \theta' \), and there remain six terms to express the value of \( m \theta + \theta' \), which we shall express by one symbol \( q \); and the equation stands thus:

\[ q = \frac{m-1}{m} \left( \frac{m^3 - 2m^2 + m + m^2 + 3m^2 + m}{an^2 + a^2n + 3rn^2 + arn} \right) \frac{e^2}{2} \]

The focal distance therefore of rays twice refracted, reckoned from the last surface, or BI, corrected for aberration, and for the thickness of the lens, is \( f = f^2 \frac{ma}{\varphi^2} - f^2 q \), consisting of three parts, viz. \( f \), the focal distance of central rays; \( f^2 \frac{ma}{\varphi^2} \), the correction for the thickness of the lens; and \( f^2 q \), the aberration.

The preceding formula appears very complex, but is of very easy management, requiring only the preparation of the simple numbers which form the numerators of the fractions included in the parenthesis. When the incident rays are parallel, the terms vanish which have \( r \) in the denominator, so that only the first three terms are used.

We might here point out the cases which reduce the aberration expressed in the formula last referred to, to nothing; but as they can scarcely occur in the object-glass of a telescope, we omit it for the present, and proceed to the combination of two or more lenses.

Lemma 3. If AG be changed by a small quantity \( Gg \); BI suffers a change \( Ii \), and \( Gg : Ii = AG^2 : BI^2 \). For it is well known that the small angles GMg and INi are equal; and therefore their subtenses \( Gk, Ii \) are proportional to MG, NI, or to AG, AI nearly, when the aperture is moderate. Therefore we have (nearly)

\[ Gk : Ii : AG : BI, Ii : Ii = AM : BI, Gg : Gk = AG : AM, \]

Therefore \( Gg : Ii = AG^2 : BI^2 \).

Prop. III. To determine the focal distance of rays refracted by two lenses placed near to each other on a common axis.

Let AM, BN (fig. II) be the surfaces of the first lens, and CO, DP be the surfaces of the second, and let \( \beta \) be the thickness of the second lens, and \( \delta \) the interval between them. Let the radius of the anterior surface of the second lens be \( a' \), and the radius of its posterior surface be \( b' \). Let \( m' \) be to 1 as the sine of incidence to the sine of refraction in the substance of the second lens. Lastly, let \( p' \) be the principal focal distance of the second lens. Let the extreme or marginal ray meet the axis in L after passing through both It is plain that DL may be determined by means of \(a'\), \(b'\), \(m'\), \(p'\), and CI in the same manner that BI was determined by means of \(a\), \(b\), \(m\), \(p\), and AG.

The value of BI is \(f - ma \frac{f^2}{\varphi} = f^2 q\). Take from this the interval \(d\), and we have CI \(= f - ma \frac{f^2}{\varphi} - d = f^2 q\).

Let the small part \(ma \frac{f^2}{\varphi} - d = f^2 q\) be neglected for the present, and let CI be supposed \(= f\): As we formed \(f_1\) and \(q_1\) by means of \(a\), \(b\), \(m\), \(n\), and \(r\), let us now form \(f_1'\) and \(q_1'\) for the second lens, by means of \(a'\), \(b'\), \(m'\), \(n'\), \(r'\) (\(= \frac{1}{a'} - \frac{1}{b'}\)), and \(r'\); then \(f'\) will be the focal distance of a slender pencil refracted by the first surface, \(f'\) will be the focal distance of this pencil after two refractions, and \(q'\) will be the co-efficient of the aberration, neglecting the thickness and interval of the lenses.

Proceeding in this way, DL will be \(= f' - m_3 \frac{f^2}{\varphi} - f^2 q'\).

But because CI is really less than \(f\), by the quantity \(ma \frac{f^2}{\varphi} + d + f^2 q'\), we must (by Lemma 3) subtract the product of this quantity, multiplied by \(\frac{DL}{BI}\) (which is nearly \(\frac{f^2}{f_1}\)), from \(f' - m_3 \frac{f^2}{\varphi} f^2 q'\).

By this process we shall have

\[DL = f' - f'' \left( \frac{ma}{\varphi} + \frac{d}{f_1} + \frac{m_3}{\varphi} \right) - f''(q + q')\]

The first term \(f'\) of this value of DI is the focal distance of a slender pencil of central rays refracted by both lenses, neglecting their thickness and distance; the second term, \(-f'' \left( \frac{ma}{\varphi} + \frac{d}{f_1} + \frac{m_3}{\varphi} \right)\), is the correction necessary for these circumstances; and the third term, \(-f''(q + q')\), is the correction for the aperture \(2e\). And it is evident that \(q'\) is a formula precisely similar to \(q\), containing the same number of terms, and differing only by the \(m'\), \(a'\), \(n'\), and \(r'\), employed in place of \(m\), \(a\), \(n\), and \(r\).

It is also evident, that if there be a third lens, we shall obtain its focal distance by a process precisely similar to that by which we obtained DL; and so on for any number of lenses.

Thus have we obtained formulae by which the foci of rays are determined in the most general terms, and in such a manner as shall point out the connection of the curvatures, thicknesses, and distances of the lenses, with their spherical aberrations, and with the final aberration of the compound lens, and give the aberrations in separate symbols, so that we can treat them by themselves, and subject them to any conditions which may enable us to correct one of them by another.

We also see in general, that the corrections for the thickness and distance of the lenses are exhibited in terms which involve only the focal distances of central rays, and have very little influence on the aberrations, and still less on the ratio of the aberrations, of the different lenses. This is a most convenient circumstance; for we may neglect them while we are determining \(q\) and \(q'\), and in determining the ratio of the focal distances of the several lenses, on which the correction of the chromatic aberration chiefly depends. Therefore, in the construction of a compound lens for uniting the different colours, we may neglect this correction for the thickness and distance till the end of the process. When we apply it, we shall find that it chiefly affects the final focal distance, making it somewhat longer, but has hardly any influence either on the chromatic or spherical aberration. We do not hesitate to say, that the final formulæ here given are abundantly accurate, while they are vastly more manageable than those employed by Euler or D'Alembert. We have calculated trigonometrically the progress of the rays through one of the glasses, which will be given as an example, giving it a very extravagant aperture, that the errors of the formulæ might be very remarkable. We found the real aberration exceed the aberration assigned by the formula by no more than \(\frac{1}{5}\)th part, a difference which is quite insignificant. The process here given derives its simplicity from the frequent occurrence of harmonic proportions in all optical theorems. This enabled M. Clairaut to employ the reciprocals of the radii and distances with so much simplicity and generality.

We consider it as another advantage of M. Clairaut's method, that it gives, by the way, formulæ for the more ordinary questions in optics, which are of wonderful simplicity, and most easily remembered. The chief problems in the elementary construction of optical instruments relate to the focal distances of central rays. This determines the focal distances and arrangement of the glasses. All the rest may be called the refinement of optics; teaching us how to avoid or correct the indistinctness, the colours, and the distortions, which are produced in the images formed by these simple constructions.

Let \(m\) be to 1 as the sine of incidence to the sine of refraction; let \(a\) and \(b\) be the radii of the anterior and posterior surfaces of a lens; let \(r\) be the distance of the radiant point, or the focus of incident central rays, and \(f\) the distance of the conjugate focus; and let \(p\) be the principal focal distance of the lens, or the focal distance of parallel rays. Make \(\frac{1}{n}\) equal to \(\frac{1}{a} - \frac{1}{b}\); let the same letters \(a'\), \(b'\), \(r'\), &c. express the same things for a second lens; and \(a''\), \(b''\), \(r''\), &c. express them for a third; and so on. Then we have

\[\frac{1}{f} = \frac{m-1}{n} + \frac{1}{r}; \quad \frac{1}{f'} = \frac{m'-1}{n'} + \frac{1}{r'}; \quad \frac{1}{f''} = \frac{m''-1}{n''} + \frac{1}{r''}.\]

Therefore, when the incident light is parallel, and \(r\) infinite, we have \(\frac{1}{p} = \frac{m-1}{n} + \frac{1}{r}; \quad \frac{1}{p'} = \frac{m'-1}{n'} + \frac{1}{r'}; \quad \frac{1}{p''} = \frac{m''-1}{n''} + \frac{1}{r''}.\)

And when several lenses are contiguous, so that their intervals may be neglected, and therefore \(\frac{1}{f}\) belonging to the first lens, becomes \(\frac{1}{r}\) belonging to the second, we have

1. \(\frac{1}{r} = \frac{1}{f} = \frac{m-1}{n} + \frac{1}{r} = \frac{1}{p} + \frac{1}{r};\) 2. \(\frac{1}{r'} = \frac{1}{f'} = \frac{m'-1}{n'} + \frac{1}{r'} = \frac{1}{p'} + \frac{1}{r'};\) 3. \(\frac{1}{r''} = \frac{m''-1}{n''} + \frac{1}{r''} = \frac{1}{p''} + \frac{1}{r''}.\)

Nothing can be more easily remembered than these formulæ, how numerous soever the glasses may be.

Having thus obtained the necessary analysis and formulæ, it now remains to apply them to the construction of achromatic lenses, in which it fortunately happens that the employment of several surfaces, in order to produce the union of the differently refrangible rays, enables us at the same time to employ them for correcting each other's spherical aberration.

A white or compounded ray is separated by refraction into its component coloured rays, and they are diffused over a small angular space. Thus it appears, that the glass used Telescope by Sir Isaac Newton in his experiments diffused a white ray, which was incident on its posterior surface in an angle of 30°, in such a manner that the extreme red ray emerged into air, making an angle of 50° 21' with the perpendicular; the extreme violet ray emerged in an angle of 51° 15'; and the ray which was in the confines of green and blue, emerged in an angle of 50° 48'. If the sine of the angle 30° of incidence be called 0.5, which it really is, the sine of the emergence of the red ray will be 0.77, that of the violet ray will be 0.78, and that of the intermediate ray will be 0.77½, an exact mean between the two extremes. This ray may therefore be called the mean refrangible ray, and the ratio of 77½ to 50, or of 1.55 to 1, will very properly express the mean refraction of this glass; and we have for this glass \( m = 1.55 \). The sine of refraction, being measured on a scale of which the sine of incidence occupies 100 parts, will be 154 for the red ray, 155 for the mean ray, and 156 for the violet ray. This number, or its ratio to unity, is commonly taken to represent the refractive power of the glass. There is some impropriety in this, unless we consider ratios as measured by their logarithms; for if \( m \) be 1, the substance does not refract at all. The refractive power can be properly measured only by the refraction which it produces; that is, by the change which it makes in the direction of the light, or the angle contained between the incident and refracted rays. If two substances produce such deviations always in one proportion, we should then say that their refractive powers are in that proportion. This is not true in any substances; but the sines of the angles contained between the refracted ray and the perpendicular, are always in one proportion when the angle of incidence in both substances is the same. This being a cognisable function of the real refraction, has therefore been assumed as the only convenient measure of the refractive powers. Although it is not strictly just, it answers extremely well in the most usual cases in optical instruments. The refractions are moderate, and the sines are very nearly as the angles contained between the rays and the perpendicular; and the real angles of refraction, or deflections of the rays, are almost exactly proportional to \( m - 1 \). The most natural and obvious measure of the refractive powers would therefore be \( m - 1 \). But this would embarrass some very frequent calculations; and we therefore find it best, on the whole, to take \( m \) itself for the measure of the refractive power.

The separation of the red, violet, and intervening rays, has been called dispersion; and although this arises merely from a difference of the refractive power in respect of the different rays, it is convenient to distinguish this particular modification of the refractive power by a name, and we call it the dispersive power of the refracting substance.

It is susceptible of degrees; for a piece of flint-glass will refract the light, so that when the sine of refraction of the red ray is 77, the sine of refraction of the violet ray is nearly 78½; or if the sine of refraction of the red ray, measured on a particular scale, is 154, the sine of refraction of the violet ray is 157. The dispersion of this substance, being measured by the difference of the extreme sines of refraction, is greater than the dispersion of the other glass, in the proportion of three to two.

But this alone is not a sufficient measure of the absolute dispersive power of a substance. Although the ratio of 1.54 to 1.56 remains constant, whatever the real magnitude of the refractions of common glass may be, and though we therefore say that its dispersive power is constant, we know that by increasing the incidence and the refraction, the absolute dispersion is also increased. Another substance shows the same properties, and in a particular case may produce the same dispersion; yet it has not for this sole reason the same dispersive power. If indeed the incidence and the refraction of the mean ray be also the same, the dispersive power cannot be said to differ; but if the incidence and the refraction of the mean ray be less, the dispersive power must be considered as greater, though the actual dispersion be the same; because if we increase the incidence till it becomes equal to that in the common glass, the dispersion will now be increased. The proper way of conceiving the dispersion therefore is, to consider it as a portion of the whole refraction; and if we find a substance making the same dispersion with half the general refraction, we must say that the dispersive quality is double, because by making the refraction equal, the dispersion will really be double.

If therefore we take \( dm \) as a symbol of the separation of the extreme rays from the middle ray, \( \frac{dm}{m-1} \) is the natural measure of the dispersive power.

It is not unusual for optical writers to take the whole separation of the red and violet rays for the measure of the dispersive power, and to compare this with the refracting power with respect to one of the extreme rays. But it is surely better to consider the mean refraction as the measure of the refracting power; and the deviation of either of the extremes from this mean is a proper enough measure of the dispersion, being always half of it. It is attended with this convenience, that being introduced into our computations as a quantity infinitely small, and treated as such for the ease of computation, while it is really a quantity of sensible magnitude, the errors arising from this supposition are diminished greatly by taking one half of the deviation and comparing it with the mean refraction. This method has, however, this inconvenience, that it does not exhibit at once the refractive power in all substances respecting any particular colour of light; for it is not the ray of any particular colour that suffers the mean refraction. In common glass it is the ray which is in the confines of the yellow and blue, in flint-glass it is nearly the middle blue ray, and in other substances it is a different ray. These circumstances appear plainly in the different proportions of the colours of the prismatic spectrum exhibited by different substances. This will be considered afterwards, being a great bar to the perfection of achromatic instruments.

The way in which an achromatic lens is constructed is, to make use of a contrary refraction of a second lens to destroy the dispersion or spherical aberration of the first.

The first purpose will be answered if \( \frac{dm}{n} \) be equal to \( \frac{dm'}{n'} \). For, in order that the different coloured rays may be collected into one point by two lenses, it is only necessary that \( \frac{1}{f} \), the reciprocal of the focal distance of rays refracted by both, may be the same for the extreme and mean rays, that is, that \( \frac{m + dm - 1}{n} + \frac{m' + dm' - 1}{n'} + \frac{1}{r} \) be of the same value with \( \frac{m - 1}{n} + \frac{m' - 1}{n'} + \frac{1}{r} \), which must happen if \( \frac{dm}{n} + \frac{dm'}{n'} \) be \( = 0 \), or \( \frac{dm}{n} = - \frac{dm'}{n'} \). This may be seen in another way, more comprehensible by such as are not versant in these discussions. In order that the extreme colours which are separated by the first lens may be rendered parallel by the second, we have shown already that \( n \) and \( n' \) are proportional to the radii of the equivalent isosceles lenses, being the halves of these radii. In these small refractions they are therefore inversely proportional to the angles formed by the surfaces Telescope at the edges of the lenses; \( n' \) may therefore be taken for the angle of the first lens, and \( n \) for that of the second.

Now the small refraction by a prism whose angle (also small) is \( n' = (m - 1)n \). The dispersive power being now substituted for the refractive power, we have for this refraction of the prism \( dm \times n' \). This must be destroyed by the opposite refraction of the other prism \( dm' \times n \).

Therefore \( dm \times n' = dm' \times n \), or \( \frac{dm}{n} = \frac{dm'}{n'} \). In like manner, this effect will be produced by three lenses if \( \frac{dm}{n} + \frac{dm'}{n'} + \frac{dm''}{n''} \) be \( = 0 \), &c.

Lastly, the errors arising from the spherical figure, which we expressed by \( R^2(q + q') \), will be corrected if \( q + q' \) be \( = 0 \). We are therefore to discover the adjustments of the quantities employed in the preceding formulae, which will insure these conditions. It will render the process more perspicuous if we collect into one view the significations of our various symbols, and the principal equations which we are to employ.

1. The ratios to unity of the sines of mean incidence in the different media are \( m, m', m'' \). 2. The ratio of the differences of the sines of the extremes \( \frac{dm}{dm'} = u \). 3. The ratio \( \frac{m-1}{m'-1} = c \). 4. The radii of the surfaces \( a, b; a', b'; a'', b'' \). 5. The principal focal distances, or the focal distances of parallel central rays \( p, p', p'' \). 6. The focal distance of the compound lens \( P \). 7. The distance of the radiant point, or of the focus of incident rays on each lens \( r, r', r'' \). 8. The focal distance of the rays refracted by each lens \( f, f', f'' \). 9. The focal distance of rays refracted by the compound lens \( F \). 10. The half breadth of the lens \( e \).

Also the following subsidiary values:

\[ \begin{align*} 1. \quad \frac{1}{a} &= \frac{1}{a'} - \frac{1}{b}; \quad \frac{1}{a'} = \frac{1}{a} - \frac{1}{b'}; \quad \frac{1}{b'} = \frac{1}{a'} - \frac{1}{b''}; \\ 2. \quad q &= \frac{m-1}{m} \left( \frac{m^2}{n^2} - \frac{2m^2 + m}{an^2} + \frac{m + 2}{a^2n} + \frac{3m^2 + m}{rn^2} - \frac{4(m+1)}{arn} + \frac{3m + 2}{rn^2} \right) \frac{e^2}{2}. \\ \end{align*} \]

And \( q' \) and \( q'' \) must be formed in the same manner from \( m', a', n', r' \); and from \( m'', a'', n'', r'' \), as \( q \) is formed from \( m, a, n, r \).

3. Also because in the case of an object-glass, \( r \) is infinitely great, the last term \( \frac{1}{r} \) in all the values of \( \frac{1}{f}, \frac{1}{f'}, \frac{1}{f''}, \frac{1}{r} \), will vanish, and we shall also have \( F = P \).

Therefore in a double object-glass \( \frac{1}{P} = \frac{m-1}{n'} + \frac{m-1}{n} \)

\[ = \frac{1}{P} + \frac{1}{P'} \]

And in a triple object-glass \( \frac{1}{P} = \frac{m-1}{n''} + \frac{m-1}{n'} + \frac{m-1}{n} \)

\[ = \frac{1}{P''} + \frac{1}{P'} + \frac{1}{P} \]

Also, in a double object-glass, the correction of spherical aberration requires \( q + q' = c \).

And a triple object-glass requires \( q + q' + q'' = v \).

For the whole error is multiplied by \( F^2 \), and by \( \frac{1}{2} e^2 \); and therefore the equation which corrects this error may be divided by \( F^2 \frac{1}{2} e^2 \).

The equation in the 14th line from the bottom of the column, giving the value of \( q, q', q'' \), may be much simplified as follows: In the first place, they may be divided by \( m, m', m'' \), by applying them properly to the terms within the parenthesis, and expunging them from the denominator of the general factors \( \frac{m-1}{m}, \frac{m'-1}{m'}, \frac{m''-1}{m''} \). This does not alter the values of \( q, q', q'' \). In the second place, the whole equations may afterwards be divided by \( m'-1 \).

This will give the values of \( \frac{q}{m'-1}, \frac{q'}{m'-1}, \frac{q''}{m'-1} \), which will still be equal to nothing if \( q + q' + q'' \) be equal to nothing.

This division reduces the general factor \( \frac{m'-1}{m'} \) of \( q' \) to \( \frac{1}{m'} \). And in the equation for \( q \) we obtain, in place of the general factor \( \frac{m-1}{m} \), the factor \( \frac{m-1}{m'-1} \), or \( c \). This will also be the factor of the value of \( q'' \) when the third lens is of the same substance with the first, as is generally the case. And, in the third place, since the rays incident on the first lens are parallel, all the terms vanish from the value of \( q \) in which \( \frac{1}{r} \) is found, and there remain only the first three, viz., \( \frac{m^2}{n^2} - \frac{2m^2 + m}{an^2} + \frac{m + 2}{a^2n} \).

Performing these operations, we have

\[ \begin{align*} \frac{q}{m'-1} &= c \left( \frac{m^2}{n^2} - \frac{2m^2 + m}{an^2} + \frac{m + 2}{a^2n} \right) \frac{e^2}{2} \\ \frac{q'}{m'-1} &= \left( \frac{m^2}{n'^2} - \frac{2m^2 + 1}{an'^2} + \frac{m + 2}{a'^2n'^2} + \frac{3m^2 + 1}{rn'^2} - \frac{(4m + 1)}{arn'^2} + \frac{3m + 2}{rn'^2} \right) \frac{e^2}{2} \\ \frac{q''}{m'-1} &= c \left( \frac{m^2}{n''^2} - \frac{2m^2 + 1}{an''^2} + \frac{m + 2}{a''^2n''^2} + \frac{3m^2 + 1}{rn''^2} - \frac{(4m + 1)}{arn''^2} + \frac{3m + 2}{rn''^2} \right) \frac{e^2}{2} \end{align*} \]

Let us now apply this investigation to the construction of an object-glass; and we shall begin with a double lens.

Construction of a Double Achromatic Object-glass.

Here we have to determine four radii, \( a, b, a', b' \). Make \( n = 1 \). This greatly simplifies the calculus, by exterminating \( n \) from all the denominators. This gives for the equation \( \frac{dm}{n} + \frac{dm'}{n'} = 0 \), the equation \( dm + \frac{dm'}{n'} = 0 \), or \( dm = -\frac{dm'}{n'} \) and \( \frac{1}{n'} = -\frac{dm}{dm'} = -u \). Also we have \( r' \), the focal distance of the light incident on the second lens, the same with the principal focal distance \( p \) of the first lens (neglecting the interval, if any). Now \( \frac{1}{p'} = \frac{m-1}{n} \), which in the present case is \( = m - 1 \). Also \( \frac{1}{p} \) is \( = -u \) (\( m' - 1 \)), and \( \frac{1}{p} = m - 1 - u \) (\( m' - 1 \)) \( = u' \).

Make these substitutions in the values of \( \frac{q}{m'-1} \) and \( \frac{q'}{m'-1} \) and we obtain the following equation. In this case \( \frac{1}{a'} = \frac{1}{b} = \frac{1}{a} - 1 \). For because \( \frac{1}{n} = \frac{1}{a} - \frac{1}{b} \) and \( n = 1 \), we have \( 1 + \frac{1}{b} = \frac{1}{a} \) and \( \frac{1}{b} = \frac{1}{a} - 1 \). Therefore, in our final equation, put \( \frac{1}{a^2} = \frac{1}{a} - \frac{2}{a} + 1 \). Therefore, in our final equation, put \( \frac{1}{a^2} = \frac{1}{a} - \frac{2}{a} + 1 \) in place of \( \frac{1}{a^2} \) and \( \frac{1}{a} - 1 \) in place of \( \frac{1}{a^2} \) and it becomes \( \frac{A - C}{a^2} - \frac{B + D - 2C}{a} + E + D - C = 0 \).

Thus have we arrived at a quadratic equation, where \( \frac{1}{a} \) is the unknown quantity. It has the form \( px^2 + qx + r = 0 \), where \( p = A - C \), \( q = 2C - B - D \), \( r = E + D - C \) and \( x = \frac{1}{a} \).

Divide the equation by \( p \), and we have \( x^2 + \frac{q}{p}x + \frac{r}{p} = 0 \). Make \( s = \frac{q}{p} \) and \( t = \frac{r}{p} \), and we have \( x^2 + sx + t = 0 \). This gives us finally \( \frac{1}{a} \), or \( x = -\frac{1}{2}s \pm \sqrt{\frac{1}{4}s^2 - t} \).

This value of \( \frac{1}{a} \) is taken from a scale of which the unit is half the radius of the isosceles lens which is equivalent to the first lens, or has the same focal distance with it. We must then find (on the same scale) the value of \( b \), viz. \( \frac{1}{a} - 1 \), which is also the value of \( a' \). Having obtained \( a' \), we must find \( b' \) by means of the equation \( \frac{1}{n'} = \frac{1}{a'} - \frac{1}{b'} \) and therefore \( \frac{1}{b} = \frac{1}{a'} - \frac{1}{n'} \). But \( \frac{1}{n'} = u \). Therefore \( \frac{1}{b} = \frac{1}{a'} + u = \frac{1}{a} + u - 1 \).

Thus is our object-glass constructed; and we must determine its focal distance, or its reciprocal \( \frac{1}{P} \). This is \( m - 1 - u(m' - 1) \).

All these radii and distances are measured on a scale of which \( n \) is the unit. But it is more convenient to measure everything by the focal distance of the compound object-glass. This gives us the proportion which all the distances bear to it. Therefore, calling \( P \) unity, in order to obtain \( \frac{1}{a} \) on this scale, we have only to state the analogy \( m - 1 - u(m' - 1) : 1 = \frac{1}{a} : \frac{1}{A} \), and \( A \) is the radius of our first surface measured on a scale of which \( P \) is the unit.

If, in the formula which expresses the final equation for \( \frac{1}{a'} \) the value of \( t \) should be positive, and greater than \( \frac{1}{4}s^2 \), the equation has imaginary roots; and it is not possible, with the glasses employed and the conditions assumed, to correct both the chromatic and spherical aberrations.

If \( t \) is negative and equal to \( \frac{1}{4}s^2 \), the radical part of the value is \( = 0 \); and \( \frac{1}{a} = -\frac{1}{2}s \). But if it be negative or positive, but less than \( \frac{1}{4}s^2 \), the equation has two real roots, which will give two constructions. That is to be preferred which gives the smallest curvature of the surfaces; because, since in our formulae which determine the spherical aberration, some quantities are neglected, these are always greater when a large arch (that is, an arch of many de- No radius should be admitted which is much less than one third of the focal distance.

This process will be made plain by an example.

Experiments have shown, that in common crown-glass the sine of incidence is to the sine of refraction as 1:526 is to 1, and that in the generality of flint-glass it is as 1:604 to 1. Also that \( \frac{dm}{dm'} = 0.6054 = u \). Therefore \( m - 1 = 0.526; m' - 1 = 0.604; c = \frac{m - 1}{m' - 1} = 0.87086 \).

By these numbers we can compute the co-efficients of our final equation. We shall find them as follows:

\[ A = 2.012, B = 3.529, C = 1.360, D = -0.526, E = 1.8659. \]

The general equation (p. 155, col. 1, line 16), when subjected to the assumed coincidence of the internal surfaces, is

\[ \frac{A}{a^3} - \frac{B}{a} + \frac{D}{a^2} - \frac{2C}{a} + E + D - C = 0. \]

\( A - C = 0.652, B + D - 2C = 0.283, \) and \( E + D - C = 0.020 \); and the equation with numerical co-efficients is

\[ \frac{0.652}{a^3} - \frac{0.283}{a} - 0.020 = 0, \]

which corresponds to the equation \( px^2 + qx + r = 0 \). We must now make \( s = \frac{q}{p} = \frac{0.283}{0.652} = 0.434 \), and \( t = \frac{r}{p} = \frac{0.02}{0.652} = 0.0307 \).

This gives us the final quadratic equation

\[ \frac{1}{a^3} - \frac{0.434}{a} - 0.0307 = 0. \]

To solve this, we have \( \frac{1}{a} = 0.217 \), and \( \frac{1}{a^2} = 0.0471 \). From this take \( t \), which is \( -0.0307 \) (that is, \( 0.0471 + 0.0307 \)), and we obtain \( 0.0778 \), the square root of which is \( 0.2789 \). Therefore, finally, \( \frac{1}{a} = 0.2170 \pm 0.2789 \), which is either \( 0.4959 \) or \( -0.0619 \). It is plain that the first must be preferred, because the second gives a negative radius, or makes the first surface of the crown-glass concave. Now as the convergence of the rays is to be produced by the crown-glass, the other surface must become very convex, and occasion great errors in the computed aberration. We therefore retain \( 0.4959 \) for the value of \( \frac{1}{a} \), and \( a = \frac{1}{0.4959} = 2.0166 \).

To obtain \( b \), use the equation \( \frac{1}{b} = \frac{1}{a} - 1 \), which gives \( \frac{1}{b} = -0.5041 \), and therefore a convex surface. Therefore \( b = \frac{1}{-0.5041} = 1.9837 \).

\( a' \) is the same with \( b \), and \( \frac{1}{a'} = -0.5041 \).

To obtain \( b' \), use the equation \( \frac{1}{b'} = \frac{1}{a'} + u \). Now \( u = 0.6054 \), and \( \frac{1}{a'} = -0.5041 \). The sum of these is \( 0.1013 \); and since it is positive, the surface is concave. \( b = \frac{1}{-0.1013} = 9.872 \).

Lastly, \( \frac{1}{P} = m - 1 - u(m' - 1) = 0.1603 \), and \( P = \frac{1}{0.1603} = 0.2383 \).

Now to obtain all the measures in terms of the focal distance \( P \), we have only to divide the measures already found by \( 0.2383 \), and the quotients are the measures wanted.

Therefore \( a = \frac{2.0166}{0.2383} = 0.32325 \),

If it be intended that the focal distance of the object-glass shall be any number \( n \) of inches or feet, we have only to multiply each of the above radii by \( n \), and we have their lengths in inches or feet.

Thus we have completed the investigation of the construction of a double object-glass. Although this was intricate, the final result is abundantly simple for practice, especially with the assistance of logarithms. The only troublesome thing is the preparation of the numerical co-efficients \( A, B, C, D, E \) of the final equation. Strict attention must also be paid to the positive and negative signs of the quantities employed.

We might propose other conditions. Thus it is natural to prefer for the first or crown-glass lens such a form as shall give it the smallest possible aberration. This will require a small aberration of the flint-glass to correct it. But a little reflection will convince us that this form will not be good. The focal distance of the crown-glass must not exceed one third of that of the compound glass; these two being nearly in the proportion of \( dm' : dm \). Therefore if this form be adopted, and \( a \) be made about \( \frac{1}{3} \)th of \( b \), it will not exceed \( \frac{1}{3} \)th of \( P \). Therefore, although we may produce a most accurate union of the central and marginal rays by opposite aberrations, there will be a considerable aberration of some rays which are between the centre and the margin.

It is absolutely impossible to collect into one point the whole rays (though the very remotest rays are united with the central rays), except in a very particular case, which cannot obtain in an object-glass; and the small quantities which are neglected in the formula which we have given for the spherical aberration, produce errors which do not follow any proportion of the aperture that can be expressed by an equation of a manageable form. When the aperture is very large, it is better not to correct the aberration for the whole aperture, but for about \( \frac{1}{3} \)ths of it. When the rays corresponding to this distance are made to coincide with the central rays by means of opposite aberrations, the rays which are beyond this distance will be united with some of those which are nearer to the centre, and the whole diffusion will be considerably diminished. Dr Smith has illustrated this in a very perspicuous manner in his theory of his Catoptric Microscope.

But although we cannot adopt this form of an object-glass, there may be other considerations which may lead us to prefer some particular form of the crown-glass, or of the flint-glass. We shall therefore adopt our general equation

\[ \frac{A}{a^3} - \frac{B}{a} - \frac{C}{a^2} - \frac{D}{a} + E = 0 \]

to this condition.

Therefore let \( h \) express this selected ratio of the two radii of the crown-glass, making \( \frac{a}{b} = h \) (remembering always that \( a \) is positive and \( b \) negative in the case of a double convex, and \( h \) is a negative number).

With this condition we have \( \frac{1}{b} = \frac{h}{a} \). But when we make \( a \) the unit of our formula of aberration, \( \frac{1}{b} = \frac{1}{a} - 1 \). Therefore \( 1 = \frac{1}{a} - \frac{h}{a} \) and \( \frac{1}{a} = \frac{1}{1 - h} \). Now substitute this for \( \frac{1}{a} \) in the general equation, and change all the signs (which still preserves it = 0), and we obtain \[ \frac{C}{a^2} + \frac{D}{a'} - E = \frac{A}{(1-h)^2} + \frac{B}{1-h} = 0. \]

By this equation we are to find \( \frac{1}{a} \) or the radius of the anterior surface of the flint-glass. The equation is of this form \( px^2 + qx + r = 0 \), and we must again make

\[ s = \frac{q}{p} \quad \text{and} \quad t = \frac{r}{p}. \]

Therefore \( s = \frac{D}{C} \) and \( t = \frac{1}{C} \times \left( \frac{B}{1-h} - \frac{A}{(1-h)^2} - E \right) \). Then, finally,

\[ \frac{1}{a} = -\frac{1}{2} s = \sqrt{\frac{1}{4}s^2 - t}. \]

It may be worth while to take a particular case of this condition. Suppose the crown-glass to be of equal convexities on both sides. This has some advantages: we can tell with precision whether the curvatures are precisely equal, by measuring the focal distance of rays reflected back from its posterior surface. These distances will be precisely equal. Now it is of the utmost importance in the construction of an object-glass which is to correct the spherical aberration, that the forms be precisely such as are required by our formulæ.

In this case of a lens equally convex on both sides \( \frac{1}{a} = -\frac{1}{b} = \frac{1}{2} \). Substitute this value for \( \frac{1}{a} \) in the general equation

\[ \frac{A}{a^2} - \frac{B}{a} - \frac{C}{a^2} - \frac{D}{a} + E = 0, \]

and then \( \frac{A}{a^2} = \frac{A}{4} \); \( \frac{B}{a} \) becomes \( \frac{B}{2} \). Now change all the signs, and we have

\[ \frac{C}{a^2} + \frac{D}{a} - E = \frac{A}{4} + \frac{B}{2} = 0, \]

by which we are to find \( a' \). This in numbers is

\[ \frac{1}{a'} = \frac{0.360}{0.526} - \frac{0.6044}{0.360} = 0.526 - 0.6044 = 0. \]

Then \( s = \frac{0.526}{0.360} = 0.3867 \), and \( t = \frac{-0.6044}{0.360} = -0.4444 \). Then \( -\frac{1}{2} s = 0.1933; \quad \frac{1}{4}s^2 = 0.0374; \quad \text{and} \quad \sqrt{\frac{1}{4}s^2 - t} = 0.6941; \]

so that \( \frac{1}{a'} = 0.1933 = 0.6941 \). This gives two real roots, viz. 0.8874, and -0.5008. If we take the first, we shall have a convex anterior surface for the flint-glass, and consequently a very deep concave for the posterior surface. We therefore take the second or negative root -0.5008.

We find \( \frac{1}{b'} \) as before, by the equation \( \frac{1}{b'} = \frac{1}{a'} + u = 0.1046 \), which will give a large value of \( b' \).

We had \( \frac{1}{a} = \frac{1}{2} \)

and \( \frac{1}{b} = -\frac{1}{2} \)

and \( \frac{1}{P} \) is the same as in the former case, viz. 0.1603.

Having all these reciprocals, we may find \( a, b, a', b', \) and \( P \); and then dividing them by \( P \), we obtain finally

\[ a = 0.3206, \quad b = -0.3206, \quad a' = -0.3201, \quad b' = 1.533, \quad P = 1. \]

By comparing this object-glass with the former, we may remark, that diminishing \( a \) a little increases \( b \), and in this respect improves the lens. It indeed has diminished \( b' \), but this being already considerable, no inconvenience attends the diminution. But we learn, at the same time, that the advantage must be very small; for we cannot diminish \( a \) much more, without making it as small as the smallest radius of the object-glass. This proportion is therefore very near the maximum, or best possible; and we know that in such cases even considerable changes in the radii will make but small changes in the result: for these reasons we are disposed to give a strong preference to the first construction, on account of the other advantages which we showed to attend it.

As another example, we may take a case which is very nearly the general practice of the London artists. The radius of curvature for the anterior surface of the convex crown-glass is \( \frac{3}{4} \)ths of the radius of the posterior surface, so that \( h = \frac{3}{4} \). This being introduced into the determinate equation, gives

\[ a = 0.2938, \quad a' = -0.3443, \quad b = -0.3526, \quad b' = 1.1474. \]

As another condition, we may suppose that the second or flint-glass is of a determined form.

This case is solved much in the same manner as the former. Taking \( h \) to represent the ratio of \( a' \) and \( b' \), we have \( \frac{1}{a'} = \frac{1}{1-h} \). This value being substituted in the general equation

\[ \frac{A}{a^2} - \frac{B}{a} - \frac{C}{a^2} - \frac{D}{a} + E = 0, \]

gives us

\[ \frac{A}{a^2} - \frac{B}{a} + E - \frac{C}{(1-h)^2} - \frac{D}{1-h} = 0. \]

This gives for the final equation

\[ x^2 + sx + t = 0, \quad s = \frac{B}{A}, \quad \text{and} \quad t = \frac{1}{A} \times \left( \frac{C}{(1-h)^2} - \frac{D}{1-h} \right), \]

and \( \frac{1}{a} = -\frac{1}{2} s = \sqrt{\frac{1}{4}s^2 - t} \).

We might here take the particular case of the flint-glass being equally concave on both sides. Then, because \( \frac{1}{a'} = -u \), and in the case of equal concavities \( \frac{2}{a'} = \frac{1}{a'} = -u \), it is sufficient to put \( -\frac{1}{2} u \) for \( \frac{1}{a'} \). This being done, the equation becomes

\[ \frac{A}{a} - \frac{B}{a} + \frac{Cu^2}{4} + \frac{Du}{2} + E = 0. \]

This gives \( s = \frac{B}{A} \), and \( t = \frac{1}{A} \cdot \left( \frac{4Du - 2Cu^2}{8} + E \right) \).

We imagine that these cases are sufficient for showing the management of the general equation; and the example of the numerical solution of the first case affords instances of the only niceties which occur in the process, viz. the proper employment of the positive and negative quantities.

We have oftener than once observed, that the formula is not perfectly accurate, and that in very large apertures errors will remain. It is proper, therefore, when we have obtained the form of a compound object-glass, to calculate trigonometrically the progress of the light through it; and if we find a considerable aberration, either chromatic or spherical, remaining, we must make such changes in the curvatures as will correct it. We have done this for the first example; and we find, that if the focal distance of the compound object-glass be 100 inches, there remains of the spherical aberration nearly \( \frac{1}{20} \)th of an inch, and the aberration of colour is over corrected above \( \frac{1}{4} \)th of an inch. The first aberration has been diminished about six times, and the other about thirty times. Both of the remaining errors will be diminished by increasing the radius of the inner surfaces. This will diminish the aberration of the crown-glass, and will diminish the dispersion of the flint. It is evident to any person conversant with optical discussions that we shall improve the correction of the spherical aberration by diminishing the refractions. If we employ two lenses for producing the convergency of the rays to a real focus, we shall reduce the aberration to \( \frac{1}{4} \)th. Therefore a better achromatic glass will be formed of three lenses, two of which are convex and of crown-glass. The refraction being thus divided between them, the aberrations are lessened. There is no occasion to employ two concave lenses of flint-glass; there is even an advantage in using one. The aberration being considerable, less of it will serve for correcting the aberration of the crown-glass, and therefore such a form may be selected as has little aberration. Some light is indeed lost by these two additional surfaces; but this is much more than compensated by the greater apertures which we can venture to give when the curvature of the surface is so much diminished. We proceed therefore to

The Construction of a Triple Achromatic Object-glass.

It is plain that there are more conditions to be assumed before we can render this a determinate problem, and that the investigation must be more intricate. At the same time it must give us a much greater variety of constructions, in consequence of our having more conditions necessary for giving the equation this determinate form. Our limits will not allow us to give a full account of all that may be done in this method. We shall therefore content ourselves with giving one case, which will sufficiently point out the method of proceeding. We shall then give the results in some other eligible cases, as rules to artists by which they may construct such glasses.

Let the first and second glasses be of equal curvatures on both sides; the first being a double convex of crown-glass, and the second a double concave of flint-glass.

Still making \( n \) the unit of our calculus, we have in the first place \( a = -b = -a' = b' \). Therefore \( \frac{1}{a'} - \frac{1}{b'} = -\left( \frac{1}{a} - \frac{1}{b} \right) \), or \( \frac{1}{n'} = -\frac{1}{n} = -1 \). Therefore the equation \( \frac{m}{n'} + \frac{m'}{n'} = 0 \) becomes \( u - 1 + \frac{u}{n'} = 0 \), or \( \frac{1}{n'} = \frac{1}{u} - 1 \).

Let us call this value \( u' \).

We have \( \frac{1}{p'} = m - 1 \); \( \frac{1}{p''} = -(m' - 1) \); \( \frac{1}{p'''} = u'(m - 1) \);

\( \frac{1}{p} = \frac{1}{p'} + \frac{1}{p''} + \frac{1}{p'''} = m - m' + u'(m - 1) \). And if we make \( m' - m = C \), we shall have \( \frac{1}{p} = -C + u'(m - 1) \).

Also \( \frac{1}{r'} = m - 1 \); \( \frac{1}{r''} = m - 1 - (m' - 1) = m - m' = -C' \).

The equality of the two curvatures of each lens gives \( \frac{1}{a} = \frac{1}{2n} \). Therefore \( \frac{1}{a} = -\frac{1}{b} = -\frac{1}{a'} = \frac{1}{b'} = \frac{1}{2} \); and \( \frac{1}{b'} = \frac{1}{a'} - \frac{1}{n'} = \frac{1}{a'} - u \).

Substituting these values in the equation (page 154, col. 9, paragraph 4), we obtain the three formulas:

1. \( cm^2 - \frac{c}{e}(2m + 1) + \frac{c(m + 2)}{4m} \) 2. \( -m^2 + \frac{1}{2}(2m' + 1) - \frac{m' + 2}{4m'} + (3m' + 1)(m - 1) \)

Now, arrange these quantities according as they are coefficients of \( \frac{1}{a'} \) and of \( \frac{1}{a''} \), or independent quantities. Let the co-efficient of \( \frac{1}{a'} \) be \( A \), that of \( \frac{1}{a''} \) be \( B \), and the independent quantity be \( C \), we have

\[ A = \frac{cu'(m + 2)}{m}, \quad B = cu'^2(2m + 1) - \frac{4ce'u'(m + 1)}{m}, \] and \[ C = cm^2 + \frac{c(m + 2)}{4m} + \frac{1}{2}(2m' + 1) + (3m' + 1)(m - 1) \] \[ + cu'm^2 + \frac{ce'u'(3m + 2)}{m} - \frac{1}{2}c(2m + 1) - m^2 - \frac{m' + 2}{4m} \]

Our equation now becomes \( \frac{A}{a'} - \frac{B}{a''} + C = 0 \).

This reduced to numbers, by computing the values of the co-efficients, is \( \frac{1}{a'} - \frac{1}{a''} = 0 \).

This, divided by 1-312, gives \( s = 0-92 \), and \( t = 0-2482 \), \( \frac{1}{s} = 0-46 \), \( \frac{1}{s'} = 0-2116 \), and \( \sqrt{\frac{1}{s} - t} = 0-6781 \).

And finally, \( \frac{1}{a'} = 0-46 = 0-6781 \).

This has two roots, viz. 0-2181 and —1-1381. The last would give a small radius, and is therefore rejected.

Now, proceeding with this value of \( \frac{1}{a'} \) and the \( \frac{1}{n'} \), we get the other radius \( b' \), and then, by means of \( u' \), we get the other radius which is common to the four surfaces. Then, by \( \frac{1}{P} = \frac{1}{p'} - c' \), we get the value of \( P \).

The radii being all on the scale of which \( u \) is the unit, they must be divided by \( P \) to obtain their value on the scale which has \( P \) for its unit. This will give us

\[ a = -b = -a' = b' = 0-530 \] \[ a' = 1-215 \] \[ b' = -0-3046 \] \[ P = 1 \]

This is not a very good form, because the last surface has too great curvature.

We thought it worth while to compute the curvatures for a case where the internal surfaces of the lenses coincide, in order to obtain the advantages mentioned on a former occasion. The form is as follows:

The middle lens is a double concave of flint-glass; the last lens is of crown-glass, and has equal curvatures on both sides. The following table contains the dimensions of the glasses for a variety of focal distances. The first column contains the focal distances in inches; the second contains the radii of the first surface in inches; the third contains the radii of the posterior surface of the first lens, and anterior surface of the second; and the fourth column has the radii of the three remaining surfaces.

| P | a | b, a' | b', a'', b' | |---|---|-------|------------| | 12 | 9-25 | 6-17 | 12-75 | | 24 | 18-33 | 12-25 | 25-5 | | 36 | 27-33 | 18-25 | 38-17 | | 48 | 36-42 | 24-33 | 50-92 | We have had an opportunity of trying glasses of this construction, and found them equal to any of the same length, although executed by an artist by no means excellent in his profession as a glass-grinder. This very circumstance gave us the opportunity of seeing the good effects of interposing a transparent substance between the glasses. We put some clear turpentine varnish between them, which completely prevented all reflection from the internal surfaces. Accordingly these telescopes were surprisingly bright; and although the roughness left by the first grinding was very perceptible by the naked eye before the glasses were put together, yet when joined in this manner it entirely disappeared, even when the glasses were viewed with a deep magnifier.

The aperture of an object-glass of this construction, of 30 inches focal distance, was 31th inches, which is considerably more than any of Mr Dollond's that we have seen.

If we should think it of advantage to make all the three lenses isosceles, that is, equally curved on both surfaces, the general equation will give the following radii:

\[ a = +0.639 \quad a' = -0.5285 \quad a'' = +0.6413 \] \[ b = -0.639 \quad b' = +0.5285 \quad b'' = -0.6413. \]

This seems a good form, having large radii.

Should we choose to have the two crown-glass lenses isosceles and equal, we must make

\[ a = +0.6412 \quad a' = -0.5287 \quad a'' = +0.6412 \] \[ b = -0.6412 \quad b' = +0.5367 \quad b'' = -0.6412. \]

This form hardly differs from the last.

Our readers will recollect that all these forms proceed on certain measures of the refractive and dispersive powers of the substances employed, which are expressed by \( m, m', dm, \) and \( dm' \); and we may be assured that the formulae are sufficiently exact, by the comparison (which we have made in one of the cases) of the result of the formulae and the trigonometrical calculation of the progress of the rays. The error was but \( \frac{1}{10} \)th of the whole, ten times less than another error, which unavoidably remains, and will be considered presently. These measures of refraction and dispersion were carefully taken; but there is great diversity, particularly in the flint-glass. We are well informed that the manufacture of this article has considerably changed of late years, and that it is in general less refractive and less dispersive than formerly. This must evidently make a change in the forms of achromatic glasses. The proportion of the focal distance of the crown-glasses to that of the flint must be increased, and this will occasion a change in the curvatures, which shall correct the spherical aberration. We examined with great care a parcel of flint-glass which an artist of this city got lately for the purpose of making achromatic object-glasses, and also some very white crown-glass made in Leith; and we obtained the following measures:

\[ m = 1.529 \quad dm = \frac{142}{219} = 0.64841. \] \[ m' = 1.578 \quad dm' = \frac{142}{219} = 0.64841. \]

We computed some forms for triple object-glasses made of these glasses, which we shall subjoin as a specimen of the variations which this change of data will occasion.

If all the three lenses are made isosceles, we have

\[ a = +0.796 \quad a' = -0.474 \quad a'' = +0.502 \] \[ b = -0.796 \quad b' = +0.474 \quad b'' = -0.502 \]

or

\[ a = 0.504 \quad a' = -0.475 \quad a'' = +0.793 \] \[ b = -0.504 \quad b' = +0.475 \quad b'' = -0.793. \]

If the middle lens be isosceles, the two crown-glass lenses may be made of the same form and focal distance, and placed the same way. This will give us

\[ a = +0.705 \quad a' = -0.475 \quad a'' = +0.705 \] \[ b = -0.547 \quad b' = +0.475 \quad b'' = -0.547. \]

This construction allows a much better form, if the measures of refraction and dispersion are the same that we used formerly. For we shall have

\[ a = +0.628 \quad a' = -0.579 \quad a'' = +0.628 \] \[ b = -0.749 \quad b' = +0.579 \quad b'' = -0.749. \]

And this is pretty near the practice of the London opticians.

We may here observe, upon the whole, that an amateur has little chance of succeeding in these attempts. The diversity of glasses, and the uncertainty of the workman's producing the very curvatures which he intends, are so great, that the object-glass turns out different from our expectation. The artist who makes great numbers acquires a pretty certain guess at the remaining error; and having many lenses intended to be of one form, but unavoidably differing a little from it, he tries several of them with the other two, and finding one better than the rest, he makes use of it to complete the set.

The great difficulty in the construction is to find the exact proportion of the dispersive powers of the crown and flint glass. The crown is pretty constant; but there are hardly two pots of flint-glass which have the same dispersive power. Even if constant, it is difficult to measure it accurately; and an error in this greatly affects the instrument, because the focal distances of the lenses must be nearly as their dispersive powers. The method of examining this circumstance, which we found most accurate, was as follows:

The sun's light, or that of a brilliant lamp, passed through a small hole in a board, and fell on another board pierced also with a small hole. Behind this was placed a fine prism A (fig. 12), which formed a spectrum RoV on a screen pierced with a small hole. Behind this was placed a prism B of the substance under examination. The ray which was refracted by it fell on the wall at D, and the distance of its illumination from that point to C, on which an unrefracted ray would have fallen, was carefully measured. This showed the refraction of that colour. Then, in order that we might be certain that we always compared the refraction of the same precise colour by the different prisms placed at B, we marked the precise position of the prism A when the ray of a particular colour fell on the prism B. This was done by an index AG attached to A, and turning with it, when we caused the different colours of the spectrum formed by A to fall on B. Having examined one prism B with respect to all the colours in the spectrum formed by A, we put another B in its place. Then bringing A to all its former positions successively, by means of a graduated arch HGK, we were certain, that when the index was at the same division of the arch, it was the very ray which had been made to pass through the first prism B in a former experiment. We did not solicitously endeavour to find the very extreme red and violet rays; because, although we did not learn the whole dispersions of the two prisms, we learned their proportions, which is the circumstance wanted in the construction of achromatic glasses. It Let us now suppose that, instead of a white spot at A, we have a prismatic spectrum AB (fig. 14), and that the prism of common glass is applied, as before, immediately behind the prism which forms the spectrum AB. We know that this will be refracted sidewise, and will make a spectrum ROYGBPC, inclined to the plane of refraction in an angle of 45°; so that drawing the perpendicular RC', we have RC' = CC.

We also know that the prism of flint-glass would refract the spectrum formed by the first prism on EHF, in such a manner that the red ray will go to R, the violet to C, and the intermediate rays to points o, y, g, b, p, v, so situated that O'o = R'O' of the other figure, Yy = R'Y' of that figure, Gg = R'G', &c. These points must therefore lie in a curve RoygbpcC, which is convex towards the axis R'C'. In like manner, we may be assured that Dr Blair's fluid will form a spectrum RoygbpcC, concave toward R'C.

Let it be observed, by the way, that this is a very good method for discovering whether a medium disperses the light in the same proportion with the prism which is employed for forming the first spectrum AB or EF. It disperses in the same or in a different proportion, according as the oblique spectrum is straight or crooked; and the exact proportion corresponding to each colour is found by measuring the ordinates of the curves RBC or R'BC.

Having formed the oblique spectrum RBC by a prism of common glass, we know that an equal prism of the same glass, placed in a contrary position, will bring back all the rays from the spectrum RBC to the spectrum AB, laying each colour on its former place.

In like manner, having formed the oblique spectrum R'BC by a prism of flint-glass, we know that another prism of flint-glass, placed in the opposite direction, will bring all the rays back to the spectrum EHF.

But having formed the oblique spectrum RBC by a prism of common glass, if we place the flint-glass prism in the contrary position, it will bring the colour R back to E, and the colour C to F; but it will not bring the colour B to H, but to a point h, such that Bh is equal to bH, and bH to hH. In like manner, the other colours will not be brought back to the straight line EHF, but to a curve EAF, forming a crooked spectrum.

In like manner, the fluids discovered by Dr Blair, when employed to bring back the oblique spectrum RBC formed by common glass, will bring its extremities back to E and F, and form the crooked spectrum EAF lying beyond EHF.

This experiment evidently gives us another method for examining the proportionality of the dispersion of different substances.

Having, by common glass, brought back the oblique spectrum formed by common glass to its natural place AB, suppose the original spectrum at AB to contract gradually (as Newton has made it to do by means of a lens), it is plain that the oblique spectrum will also contract, and so will the second spectrum at AB; and it will at last coalesce into a white spot. The effect will be equivalent to a gradual compression of the whole figure, by which the parallel lines AR and BC gradually approach, and at last unite.

In like manner, when the oblique spectrum formed by Telescope. Flint-glass is brought back to EHF by a flint-glass prism, and the figure compressed in the same gradual manner, all the colours will coalesce into a white spot.

But when flint-glass is employed to bring back the oblique spectrum formed by common glass, it forms the crooked spectrum EAF. Now let the figure be compressed. The curve EAF will be doubled down on the line HA, and there will be formed a compound spectrum Hh, quite unlike the common spectrum, being purple or claret coloured at H by the mixture of the extreme red and violet, and green edged with blue at h by the mixture of the green and blue. The fluid prisms would in like manner form a spectrum of the same kind on the other side of H.

This is precisely what is observed in achromatic object-glasses made of crown-glass and flint: for the refraction from A to R corresponds to the refraction of the convex crown-glass; and the contrary refraction from R to E corresponds to the contrary refraction of the concave flint-glass, which still leaves a part of the first refraction, producing a convergence to the axis of the telescope. It is found to give a purple or wine-coloured focus, and within this a green one, and between these an imperfect white. Dr Blair found, that when the eye-glass was drawn out beyond its proper distance, a star was surrounded by a green fringe, by the rays of the green end of the spectrum, which crossed each other within the focus; and when the eye-glass was too near the object-glass, the star had a wine-coloured fringe. The green rays were ultimately most refracted. We should expect the fringe to be of a blue colour rather than a green. But this is easily explained. The extreme violet rays are very faint, so as hardly to be sensible; therefore, when a compound glass is made as achromatic as possible to our senses, in all probability (nay certainly) these almost insensible violet rays are left out, and perhaps the extreme colours which are united are the red and the middle violet rays. This makes the green to be the mean ray, and therefore the most outstanding when the dispersions are not proportional.

Dr Blair very properly calls these spectrums, Hh and Hh', secondary spectrums, and seems to think that he is the first who has taken notice of them as indispensably necessary. But this subject was touched by Clairaut, and fully discussed by Boscovich.

The most essential service which the public has received at the hands of Dr Blair, is the discovery of fluid mediums of proper dispersive power. By composing the lenses of such substances, we are at once freed from the irregularities in the refraction and dispersion of flint-glass, from which the chemists have not been able to free it. In whatever way this glass is made, it consists of parts which differ both in refractive and dispersive power; and when taken up from the pot, these parts mix in threads, which may be disseminated through the mass in any degree of fineness. But they still retain their properties; and when a piece of flint-glass has been formed into a lens, the eye placed in its focus, sees the whole surface occupied by glistering threads or broader veins running across it. Great rewards have been offered for removing this defect, but hitherto to no purpose. We beg leave to propose the following method. Let the glass be reduced to powder, and then melted with a great proportion of alkaline salt, so as to make a liquor silicum. When precipitated from this by an acid, it must be in a state of very uniform composition. If again melted into glass, we should hope that it would be free from this defect; if not, the case seems to be desperate. But by using a fluid medium, Dr Blair was released from all this embarrassment; and he acquired another immense advantage, that of adjusting at pleasure both the refractive and dispersive powers of his lenses. In solid lenses, we do not know whether we have taken the curvatures suited to the refractions till our glass is finished; and if we have mistaken the proportions, all our labour is lost. But when fluids are used, it is enough that we know nearly the refractions. We suit our focal distances to these, and then select our curvature, so as to remove the aberration of figure, preserving the focal distances. Thus, by properly tempering the fluid mediums, we bring the lens to agree precisely with the theory, perfectly achromatic, and the aberration of figure as much corrected as is possible.

Dr Blair examined the refractive and dispersive powers of a great variety of substances, and found great varieties in their actions on the different colours. This is indeed what every well-informed naturalist would expect. There is no doubt now among naturalists about the mechanical connection of the phenomena of nature; and all are agreed that the chemical actions of the particles of matter are perfectly like in kind to the action of gravitating bodies; that all these phenomena are the effects of forces like those which we call attractions and repulsions, and which we observe in magnets and electrified bodies; that light is refracted by forces of the same kind, but differing chiefly in the small extent of their sphere of activity. We cannot derive much knowledge from what he has already published, because it was chiefly with the intention of giving a popular, though not an accurate, view of the subject. The constructions which are there mentioned are not those which he found most effectual, but those which would be most easily understood, or demonstrated by the slight theory which is contained in the dissertation; besides, the manner of expressing the difference of refrangibility, perhaps chosen for its paradoxical appearance, does not give us a clear notion of the characteristic differences of the substances examined. Those rays which are ultimately most deflected from their direction, are said to have become the most refrangible by the combination of different substances, although in all the particular refractions by which this effect is produced, they are less refracted than the violet light. We can just gather this much, that common glass disperses the rays in such a manner that the ray which is in the centre of the green and blue occupies the middle of the prismatic spectrum; but in glasses, and many other substances which are more dispersive, this ray is nearer to the ruddy extremity of the spectrum. While therefore the straight line RC' (fig. 14) terminates the ordinates Oo', Y'Y', Gg', &c., which represent the dispersion of common glass, the ordinates which express the dispersions of these substances are terminated by a curve passing through R and C', but lying below the line RC'. When therefore parallel heterogeneous light is made to converge to the axis of a convex lens of common glass, as happens at F (in fig. 8) the light is dispersed, and the violet rays have a shorter focal distance. If we now apply a concave lens of greater dispersive power, the red and violet rays are brought to one focus F'; but the green rays, not being so much refracted away from F, are left behind at F, and have now a shorter focal distance. But Dr Blair afterwards found that this was not the case with the muriate acid, and some solutions in it. He found that the ray which common glass caused to occupy the middle of the spectrum, was much nearer to the blue extremity when refracted by these fluids. Therefore a concave lens formed of such fluids which united the red and violet rays in F', refracted the green rays to F'.

Having observed this, it was an obvious conjecture, that a mixture of some of these fluids might produce a medium, whose action on the intermediate rays should have the same proportion that is observed on common glass; or that two of them might be found which formed spectra similarly divided, and yet differing sufficiently in dispersive power to enable us to destroy the dispersion by contrary refractions, without destroying the whole refraction. Dr Blair accordingly found a mixture of ammoniacal and mercurial Telescope salts, and also some other substances, which produced dispersions proportional to that of glass, with respect to the different colours.

And thus did the result of this intricate and laborious investigation correspond to his utmost wishes. He produced achromatic telescopes which seem as perfect as the thing will admit of; for he was able to give them such apertures, that the incorrigible aberration arising from the spherical surfaces becomes a sensible quantity, so as to preclude farther amplification of the eye-glasses. We have examined one of his telescopes. The focal distance of the object-glass did not exceed 17 inches, and the aperture was fully 3½ inches. We viewed some single and double stars, and some common objects, with this telescope; and found, that in magnifying power, brightness, and distinctness, it was manifestly superior to one of Mr Dollond's of 42 inches focal length. It also gave us an opportunity of admiring the dexterity of the London artists, who could work the glasses with such accuracy. We had most distinct vision of a star when using an erecting eye-piece, which made this telescope magnify more than a hundred times; and we found the field of vision as uniformly distinct as with Dollond's 42-inch telescope magnifying forty-six times. The intelligent reader must admire the nice figuring and centring of the very deep eye-glasses which are necessary for this amplification.

We now proceed to consider the eye-pieces or glasses of telescopes. The proper construction of an eye-piece is not less essential than that of the object-glass. But our limits will not allow us to treat this subject in the same detail. Our readers will find abundant information in Dr Smith's Optics concerning the eye-glasses, chiefly deduced from Huyghens's fine theory of aberration. At the same time, we must again pay Mr Dollond the merited compliment of saying, that he was the first who made any scientific application of this theory to the compound eye-piece for erecting the object. His eye-pieces of five and six glasses are very ingenious reduplications of Huyghens's eye-pieces of two glasses, and would probably have superseded all others, had not his discovery of achromatic object-glasses caused opticians to consider the chromatic dispersion with more attention, and pointed out methods of correcting it in the eye-piece without any compound eye-glasses. They have found that this may be more conveniently done with four eye-glasses, without sensibly diminishing the advantages which Huyghens showed to result from employing many small refractions instead of a lesser number of great ones. As this is a curious subject, we shall add what may be sufficient for making our readers fully acquainted with it, and content ourselves with merely mentioning the principles of the other rules for constructing an eye-piece.

Such readers as are less familiarly acquainted with optical discussions will do well to keep in mind the following consequences of the general focal theorem.

If AB (fig. 15) be a lens, R a radiant point or focus of incident rays, and a the focus of parallel rays coming from the opposite side; then,

1. Draw the perpendicular ac' to the axis, meeting the incident ray in a', and a'A to the centre of the lens. The refracted ray BF is parallel to a'A; for Ra': a'A (= Ra : aA) = RB : BF (= RA : AF), which is the focal theorem.

2. An oblique pencil BP6 proceeding from any point P which is not in the axis, is collected to the point f, where the refracted ray BF cuts the line PAf drawn from P through the centre of the lens; for Pa': a'A = PB : Bf, which is also the focal theorem.

The Galilean telescope is susceptible of so little improve-

ment that we need not employ any time in illustrating its Telescope performance.

The simple astronomical telescope is represented in fig. 16. The beam of parallel rays, inclined to the axis, is made to converge to a point G, where it forms an image of the lowest point of a very distant object. These rays decussating from G fall on the eye-glass; the ray from the lowest point B of the object-glass falls on the eye-glass at b; and the ray from A falls on a; and the ray from the centre O falls on o. These rays are rendered parallel, or nearly so, by refraction through the eye-glass, and take the direction bo', ol, ai. If the eye be placed so that this pencil of parallel rays may enter it, they converge to a point of the retina, and give distinct vision to the lowest point of the object. It appears inverted, because the rays by which we see its lowest point come in the direction which in simple vision is connected with the upper point of an object. They come from above, and therefore are thought to proceed from above. We see the point as if situated in the direction lo. In like manner, the eye placed at I sees the upper point of the object in the direction IP, and its middle in the direction IE. The proper place for the eye is I; if brought much nearer the glass, or removed much farther from it, some or the whole of this extreme pencil of rays will not enter the pupil. It is therefore of importance to determine this point. Because the eye requires parallel rays for distinct vision, it is plain that F must be the principal focus of the eye-glass. Therefore, by the common focal theorem, OF : OE = OE : OI, or OF : FE = OE : EI.

The magnifying power being measured by the magnitude of the visual angle, compared with the magnitude of the visual angle with the naked eye, we have \(\frac{OF}{OP}\) or \(\frac{OF}{OF}\) for the measure of the magnifying power. This is very nearly \(OE : EI\) or \(OF : FI\).

As the line OE, joining the centres of the lenses, and perpendicular to their surfaces, is called the axis of the telescope, so the ray OG is called the axis of the oblique pencil, being really the axis of the cone of light which has the object-glass for its base. This ray is through its whole course the axis of the oblique pencil; and when its course is determined, the amplification, the field of vision, the apertures of the glasses, are all determined. For this purpose we have only to consider the centre of the object-glass as a radical point, and trace the process of a ray from this point through the other glasses; this will be the axis of some oblique pencil.

It is evident, therefore, that the field of vision depends on the breadth of the eye-glass. Should we increase this, the extreme pencil will pass through I, because O and I are still the conjugate foci of the eye-glass. On the other hand, the angle resolved on for the extent or field of vision gives the breadth of the eye-glass.

We may here observe, by the way, that for all optical instruments there must be two optical figures considered. The first shows the progress of a pencil of rays coming from one point of the object. The various focuses of this pencil show the places of the different images, real or virtual. Such a figure is formed by the three rays Aga', The second shows the progress of the axes of the different pencils proceeding through the centre of the object-glass. The focuses of this pencil of axes show the places where an image of the object-glass is formed; and this field determines the field of vision, the apertures of the lenses, and the amplification or magnifying power. The three rays OGoI, OFEI, OHPI, form this figure.

See also fig. 24, where the progress of both sets of pencils is more diversified.

The perfection of a telescope is to represent an object in its proper shape, distinctly magnified, with a great field of vision, and sufficiently bright. But there are limits to all these qualities; and an increase of one of them, for the most part, diminishes the rest. The brightness depends on the aperture of the object-glass, and will increase in the same proportion (because \( \frac{a}{b} \) will always be to \( \frac{AB}{EF} \) in the proportion of EF to FO), till the diameter of the emergent pencil is equal to that of the pupil of the eye. Increasing the object-glass any more, can send no more light into the eye. But we cannot make the emergent pencil nearly so large as this when the telescope magnifies much; for the great aperture of the object-glass produces an indistinct image at GF, and its indistinctness is magnified by the eyeglass.

A great field of vision is incompatible with the true shape of the object; for it is not strictly true that all rays flowing from O are refracted to I. Those rays which go to the margin of the eye-glass cross the axis between E and I; and therefore they cross it at a greater angle than if they passed through I. Now had they really passed through I, the object would have been represented in its due proportions. Therefore, since the angles of the marginal parts are enlarged by the aberration of the eye-glass, the marginal parts themselves will appear enlarged, or the object appear distorted. Thus a chess-board viewed through a reading glass appears drawn out at the corners, and the straight lines are all changed into curves, as is represented in fig. 17.

The circumstance which most perceptibly limits the extent of field is the necessary distinctness. If the vision be indistinct, it is useless, and no other quality can compensate this defect. The distortion is very inconsiderable in much larger angles of vision than we can admit, and is unworthy of the attention paid to it by optical writers. They have been induced to take notice of it, because the means of correcting it in a considerable degree are attainable, and afford an opportunity of exhibiting their knowledge; whereas the indistinctness which accompanies a large field is a subject of most difficult discussion, and has hitherto baffled all their efforts to express it by any intelligible or manageable formulae.

This subject must however be considered. The image at GF of a very remote object is not a plain surface perpendicular to the axis of the telescope, but is nearly spherical, having O for its centre. If a number of pencils of parallel rays crossing each other in I fall on the eye-glass, they will form a picture on the opposite side, in the focus F. But this picture will by no means be flat, nor nearly so, but very concave towards E. Its exact form is of most difficult investigation. The elements of it are given by Dr Barrow; and we have given the chief of them in the article OPTICS, when considering the foci of infinitely slender pencils of oblique rays. Therefore it is impossible that the picture formed by the object-glass can be seen distinctly in all its parts by the eye-glass. Even if it were flat, the points G' and H (fig. 16) are too far from the eye-glass when the middle F is at the proper distance for distinct vision. When, therefore, the telescope is so adjusted that we have distinct vision of the middle of the field, in order to see the margin distinctly we must push in the eye-glass; and having so done, the middle of the field becomes indistinct. When the field of vision exceeds twelve or fifteen degrees, it is not possible by any contrivance to make it tolerably distinct all over; and we must turn the telescope successively to the different parts of the field that we may see them agreeably.

The cause of this indistinctness is, as we have already said, the shortness of the lateral foci of lateral and oblique pencils refracted by the eye-glass. The oblique pencil \( \alpha G_o \), by which an eye placed at I sees the point G of the image, is a cone of light, having a circular base on the eyeglass, of which circle \( ab \) is one of the diameters. There is a diameter perpendicular to this, which, in this figure, is represented by the point \( \alpha \). Fig. 18 represents the base of the cone as seen by an eye placed in the axis of the telescope, with the object-glass as appearing behind it. The point \( b \) is formed by a ray which comes from the lowest point B of the object-glass, and the point \( \alpha \) is illuminated by a ray from A. The point \( c \) at the right hand of the circular base of this cone of light came from the point C on the left side of the object-glass; and the light comes to \( d \) from D. Now the laws of optics demonstrate, that the rays which come through the points \( c \) and \( d \) are more convergent after refraction than the rays which come through \( a \) and \( b \). The analogies, therefore, which ascertain the foci of rays lying in planes passing through the axis, do not determine the foci of the others. Of this we may be sensible by looking through a lens to a figure on which are drawn concentric circles crossed by radii. When the telescope is so adjusted that we see distinctly the extremity of one of the radii, we shall not see distinctly the circumference which crosses the extremity with equal distinctness, and vice versa. This difference, however, between the foci of the rays which come through \( a \) and \( b \), and those which come through \( c \) and \( d \), is not considerable in the fields of vision which are otherwise admissible. But the same difference of foci obtains also with respect to the dispersion of light, and is more remarkable. Both D'Alembert and Euler have attempted to introduce it into their formulae; but they have made them useless for any practical purpose by their inextricable complication.

This must serve as a general indication of the difficulties which occur in the construction of telescopes, even although the object-glass were perfect, forming an image without the smallest confusion or distortion.

There is yet another difficulty or imperfection. The rays of the pencil \( \alpha G_o \) (fig. 16), when refracted through the eyeglass, are also separated into their component colours. The edge of the lens must evidently perform the office of a prism, and the white ray \( G_o \) will be so dispersed, that if \( b_i \) be the path of its red ray, the violet ray, which makes another part of it, will take such a course \( b_m \) that the angle \( r_{bm} \) will be nearly \( \frac{1}{3} \)th of \( G_o \). The ray \( G_a \) passing through a part of the lens whose surfaces are less inclined to each other, will be less refracted, and will be less dispersed in the same proportion very nearly. Therefore the two violet rays will be very nearly parallel when the two red rays are rendered parallel.

Hence it must happen that the object will appear bordered with coloured fringes. A black line seen near the margin on a white ground, will have a ruddy and orange border on the outside, and a blue border within; and this confusion is altogether independent on the object-glass, and is so much the greater as the visual angle \( bIE \) is greater.

Such are the difficulties. They would be unsurmountable, were it not that some of them are so connected that, to a certain extent, the diminution of one is accompanied... Telescope by a diminution of the other. What are called the caustic curves are the geometrical loci of the foci of infinitely slender pencils; consequently the point G is very nearly in the caustic formed by a beam of light consisting of rays parallel to Io, and occupying the whole surface of the eyeglass, because the pencil of rays which are collected at G is very small. Any thing therefore that diminishes the mutual inclination of the adjoining rays, puts their concourse farther off. Now this is precisely what we want; for the point G of the image formed by the object-glass is already beyond the focus of the oblique slender pencil of parallel rays ia and ib; and therefore, if we could make this focus go a little farther from a and b, we shall bring it nearer to G, and obtain more distinct vision of this point of the object. Now let it be recollected, that in moderate refractions through prisms, two rays which are inclined to each other in a small angle are, after refraction, inclined to each other in the same angle. Therefore, if we can diminish the aberration of the ray ai, or ei, or bi', we diminish their mutual inclination, and consequently the mutual inclination of the rays Ga, Go, Gi', and therefore lengthen the focus, and get more distinct vision of the point G. Therefore we at once correct the distortion and the indistinctness; and this is the aim of Mr Huyghens's great principle of dividing the refractions.

The general method is as follows. Let O be the object-glass (fig. 19) and E the eye-glass of a telescope, and F their common focus, and FG the image formed by the object-glass. The proportion of their focal distances is supposed to be such as gives as great a magnifying power as the perfection of the object-glass will admit. Let BI be the axis of the emergent pencil. It is known by the focal theorem that GE is parallel to BI; therefore BGE is the whole refraction or deflection of the ray OHB from its former direction. Let it be proposed to diminish the aberrations by dividing this into two parts by means of two glasses D and e, so as to make the ultimate angle of vision bi' equal to BIE, and thus retain the same magnifying power and visible field. Let it be proposed to divide it into the parts BGC and CGE.

From G draw any line GD to the axis towards O; and draw the perpendicular DH, cutting OG in H; draw He parallel to GC, cutting GD in g; draw gf perpendicular to the axis, and ge parallel to GE; draw eb perpendicular to the axis; draw Dd parallel to GC, and dG perpendicular to the axis. Then if there be placed at D a lens whose focal distance is Dd, and another at e whose focal distance is ef, the thing is done. The ray OH will be refracted into Hb, and this into bi', parallel to BI.

The demonstration of this construction is so evident by means of the common focal theorem, that we need not repeat it, nor the reasons for its advantages. We have the same magnifying power, and the same field of vision; we have less aberration, and therefore less distortion and indistinctness; and this is brought about by a lens HD of a smaller aperture and a greater focal distance than BE. Consequently, if we are contented with the distinctness of the margin of the field with a single eye-glass, we may greatly increase the field of vision; for if we increase DH to the size of EB, we shall have a greater field, and much greater distinctness in the margin; because HD is of a longer focal distance, and will bear a greater aperture, preserving the same distinctness at the edge. On this account the glass HD is commonly called the field-glass.

It must be observed here, however, that although the distortion of the object is lessened, there is a real distortion produced in the image fig. But this, when magnified by the glass e, is smaller than the distortion produced by the glass E, of greater aperture and shorter focus, on the undistorted image GF. But because there is a distortion in the second image fig., this construction cannot be used for the telescopes of astronomical quadrants, and other graduated instruments, because then equal divisions of the micrometer would not correspond to equal angles.

But the same construction will answer in this case by taking the point D on that side of F which is remote from O (fig. 20). This is the form now employed in the telescopes of all graduated instruments.

Fig. 20.

The exact proportion in which the distortion and the indistinctness at the edges of the field are diminished by this construction, depends on the proportion in which the angle BGE is divided by GC, and is of pretty difficult investigation. But it never deviates far (never one eighth in optical instruments) from the proportion of the squares of the angles. We may, without any sensible error, suppose it in this proportion. This gives us a practical rule of easy recollection, and of most extensive use. When we would diminish an aberration by dividing the whole refraction into two parts, we shall do it most effectually by making them equal. In like manner, if we divide it into three parts by means of two additional glasses, we must make each = \( \frac{1}{3} \) of the whole, and so on for a greater number.

This useful problem, even when limited, as we have done, to equal refractions, is as yet indeterminate; that is, susceptible of an infinity of solutions; for the point D, where the field-glass is placed, was taken at pleasure; yet there must be situations more proper than others. The aberrations which produce distortion, and those which produce indistinctness, do not follow the same proportions. To correct the indistinctness, we should not select such positions of the lens HD as will give a small focal distance to be; that is, we should not remove it very far from F. Huyghens recommends the proportion of three to one for that of the focal distances of the lens HD and eb, and says that the distance De should be \( = 2Fe \). This will make \( ei = \frac{1}{3} \) of \( ef \), and will divide the whole refraction into two equal parts, as any one will readily see by constructing the common optical figure. Mr Short, the celebrated improver of reflecting telescopes, generally employed this proportion; and we shall presently see that it is a very good one.

It has already been observed that the great refractions which take place on the eye-glasses, occasion very considerable dispersions, and disturb the vision by fringing everything with colours. To remedy this, achromatic eyeglasses may be employed, constructed by the rules already delivered. This construction, however, is incomparably more intricate than that of object-glasses; for the equations must involve the distance of the radiant point, and be more complicated; and this complication is immensely increased on account of the great obliquity of the pencils.

Most fortunately the Huyghenian construction of an eyepiece enables us to correct this dispersion to a great degree of exactness. A heterogeneous ray is dispersed at H, and the red ray belonging to it falls on the lens be, at a greater distance from the centre than the violet ray coming from H. It will therefore be less refracted, ceteris paribus, by Let the compound ray OP (fig. 21) be dispersed by the lens PC, and let PV, PR be its violet and red rays, cutting the axis in g and G. It is required to place another lens RD in their way, so that the emergent rays Rr, Ve shall be parallel.

Produce the incident ray OP to Z. The angles ZPR, ZPV, are given (and RPV is nearly \( \frac{ZPR}{27} \)), and the intersections G and g with the axis. Let F be the focus of parallel red light coming through the lens RD in the opposite direction. Then (by the common optical theorem) the perpendicular \( f_g \) will cut PR in such a point \( s \), that \( f \) will be parallel to the emergent ray Rr, and to Ve. Therefore if \( f_g \) cut PV in u, and \( uf \) be drawn perpendicular to the axis, we shall have (also by the common theorem) the point \( f \) for the focus of violet rays, and DF : Df = Dz : Du = 28 : 27 nearly, or in a given ratio.

The problem is therefore reduced to this, "to draw from a point D in the line CG a line \( D_2 \), which shall be cut by the lines PR and PV in the given ratio."

The following construction naturally offers itself. Make GM : \( \varphi M \) in the given ratio, and draw MK parallel to Pg. Through any point D of CG draw the straight line PDK, cutting MK in K. Join GK, and draw \( D_2 \) parallel to KG. This will solve the problem; and, drawing \( f_g \) perpendicular to the axis, we shall have F for the focus of the lens RD for parallel red rays.

The demonstration is evident; for MK being parallel to Pg, we have GM : \( \varphi M = GK : HK = \frac{D : uD}{FD : D} \), in the ratio required.

This problem admits of an infinity of solutions, because the point D may be taken anywhere in the line CG. It may therefore be subjected to such conditions as may produce other advantages.

1. It may be restricted by the magnifying power, or by the division which we choose to make of the whole refraction which produces this magnifying power. Thus, if we have resolved to diminish the aberrations by making the two refractions equal, we have determined the angle RrD. Therefore draw GK, making the angle MGK equal to that which the emergent pencil must make with the axis, in order to produce this magnifying power. Then draw MK parallel to Pg, meeting GK in K. Then draw PK, cutting the axis in D, and \( D_2 \) parallel to GK, and \( f_g \) perpendicular to the axis. D is the place and DF the focal distance of the eye-glass.

2. Particular circumstances may cause us to fix on a particular place D, and we only want the focal distance. In this case the first construction suffices.

3. We may have determined on a certain focal distance DF, and the place must be determined. In this case let

\[ \begin{align*} GF : f_g &= 1 : \tan G, \\ f_g : fu &= 1 : m, m being = \frac{27}{26}, \end{align*} \]

then

\[ \begin{align*} GF : f_g &= \tan g : m \tan G, \\ GF - f_g : GF &= \tan g - m \tan G : \tan g, \\ Gg + f_g : GF &= \tan g - m \tan G : \tan g, \end{align*} \]

and

\[ GF = Gg + f_g \tan g - m \tan G, \]

and therefore given, and the place of F is determined; and since FD is given by supposition, D is determined.

The application of this problem to our purpose is difficult, if we take it in the most general terms; but the nature of the thing makes such limitations that it becomes very easy. In the case of the dispersion of light, the angle GPg is so small that MK may be drawn parallel to PG without any sensible error. If the ray OP were parallel to CG, then G would be the focus of the lens PC, and the point M would fall on C; because the focal distance of red rays is to that of violet rays in the same proportion for every lens, and therefore CG : Cg = DF : Df. Now, in a telescope which magnifies considerably, the angle at the object-glass is very small, and CG hardly exceeds the focal distance; and CG is to Cg very nearly in the same proportion of 28 to 27. We may therefore draw through C (fig. 22) a line CK parallel to PG; then draw GK perpendicular to the axis of the lenses, and join PK; draw KBE parallel to CG, cutting PK in B; draw BHI parallel to GK, cutting GK in H. Join HD and PK. It is evident that CG is bisected in F', and that KB = 2FD; also KH : HG = KB : BE = CD : DG. Therefore DH is parallel to CK' or to PG. But because PF = F'K', PD is = DB, and IH = HB. Therefore \( f_g \)D = HB, and FD = KB = 2FD; and FD is bisected in F'. Therefore

\[ CD = \frac{CG + FD}{2}. \]

That is, in order that the eye-glass RD may correct the dispersion of the field-glass PC, the distance between them must be equal to the half sum of their focal distances very nearly. More exactly, the distance between them must be equal to the half sum of the focal distance of the eye-glass, and the distance at which the field-glass would form an image of the object-glass. For the point G is the focus to which a ray coming from the centre of the object-glass is refracted by the field-glass.

This is a very simple solution of an important problem. Huyghens's eye-piece corresponds with it exactly. If indeed the dispersion at P is not entirely produced by the refraction, but perhaps combined with some previous dispersion, the point M (fig. 21) will not coincide with C (fig. 22), and we shall have GC to GM as the natural dispersion at P to the dispersion which really obtains there. This may destroy the equation CD = \( \frac{CG + FD}{2} \).

Thus, in a manner rather unexpected, have we freed the eye-glasses from the greatest part of the effect of dispersion. We may do it entirely by pushing the eye-glass a little nearer to the field-glass. This will render the violet rays a little divergent from the red, so as to produce a perfect picture at the bottom of the eye. But by doing so we have hurt the distinctness of the whole picture, because F is not in the focus of RD. We remedy this by This improvement cannot be applied to the construction of quadrant telescopes, such as fig. 20. Mr Ramsden has attempted it, however, in a very ingenious way, which merits a place here, and is also instructive in another respect. The field-glass HD is a plano-convex, with its plane side next the image GF. It is placed very near this image. The consequence of this disposition is, that the image GF produces a vertical image gf', which is much less convex towards the glass. He then places a lens on the point C, where the red ray would cross the axis. The violet ray will pass on the other side of it. If the focal distance of this glass be fc, the vision will be distinct and free from colour. It has, however, the inconvenience of obliging the eye to be close to the glass, which is very troublesome.

This would be a good construction for a magic lanthorn, or for the object-glass of a solar microscope, or indeed of any compound microscope.

We may presume that the reader is now pretty familiar with the different circumstances which must be considered in the construction of an eye-piece, and proceed to consider those which must be employed to erect the object.

This may be done by placing the lens which receives the light from the object-glass in such a manner that a second image (inverted with respect to the first) may be formed beyond it, and this may be viewed by an eye-glass. Such a construction is represented in fig. 23. But, besides many other defects, it tinges the object prodigiously with colour. The ray cd is dispersed at d into the red ray dr and the violet dv, v being farther from the centre than r; the refracted ray vr' crosses rr', both by reason of spherical aberration and its greater refrangibility.

But the common day-telescope, invented by F. Rhetta, has in this respect greatly the advantage of the one now described. The rays of compound light are dispersed at two points. The violet ray in its course falls without the red ray, but is accurately collected with it at a common focus, as we shall demonstrate by and by. Since they cross each other in the focus, the violet ray must fall within the red ray, and be less refracted than if it had fallen on the same point with the red ray. Had it fallen there, it would have separated from it; but by a proper diminution of its refraction, it is kept parallel to it, or nearly so. And this is one excellence of this telescope; when constructed with three eye-glasses perfectly equal, the colour is sensibly diminished, and by using an eye-glass somewhat smaller it may be removed entirely. We say no more of it at present, because we shall find its construction included in another, which is still more perfect.

It is evident at first sight that this telescope may be improved by substituting for the eyeglass the Huyghenian double eyeglass, or field-glass and eye-glass represented in figs. 19 and 20, and that the first of these may be improved and rendered achromatic. This will require the two glasses to be increased from their present dimensions to the size of a field-glass suited to the magnifying power of the telescope, supposing it an astronomical telescope. Thus we shall have a telescope of four eye-glasses. The first three will be of a considerable focal distance, and two of them will have a common focus at b. But this is considerably different from the eye-pieces of four glasses which are now used, and are far better. We are indebted for them to Mr Dollond, who was a mathematician as well as an artist, and in the course of his research discovered resources which had not been thought of. He had not then discovered the achromatic object-glass, and was busy in improving the eye-glasses by diminishing their spherical aberration. His first thought was to make the Huyghenian addition at both the images of the day-telescope. This suggested to him the following eye-piece of five glasses.

Fig. 24 represents this eye-piece, but there is not room for the object-glass at its proper distance. A pencil of rays coming from the upper point of the object is made to converge (by the object-glass) to G, where it would form a picture of that part of the object. But it is intercepted by the lens Aa, and its axis is bent towards the axis of the telescope in the direction ah. At the same time, the rays which converged to G converge to g, and there is formed an inverted picture of the object at gf. The axis of the pencil is again refracted at b, crosses the axis of the telescope in H, is refracted again at c, at d, and at e, and at last crosses the axis in I. The rays of this pencil, diverging from g, are made less diverging, and proceed as if they came from g', in the line Bgg'. The lens eC causes them to converge to g', in the line G'Cg'. The lens dD makes them converge still more to G", and there they form an erect picture G"Fe"; diverging from G", they are rendered parallel by the refraction at e.

At H the rays are nearly parallel. Had the glass Bb been a little farther from A, they would have been accurately so, and the object-glass, with the glasses A and B, would have formed an astronomical telescope with the Huyghenian eye-piece. The glasses C, D, and E, are intended merely for bending the rays back again till they again cross the axis in I. The glass C tends chiefly to diminish the great angle BHB; and then the two glasses D and E are another Huyghenian eye-piece.

The art in this construction lies in the proper adjustment of the glasses, so as to divide the whole bending of the pencil pretty equally among them, and to form the last image in the focus of the eye-glass, and at a proper distance from the other glass. Bringing B nearer to A would bend the pencil more to the axis. Placing C farther from B would do the same thing; but this would be accompanied with more aberration, because the rays would fall at a greater distance from the centres of the lenses. The greatest bending is made at the field-glass D; and we imagine that the There is an image formed at H of the object-glasses, and the whole light passes through a small circle in this place. It is usual to put a plate here pierced with a hole which has the diameter of this image. A second image of the object-glass is formed at I, and indeed wherever the pencils cross the axis. A lens placed at H makes no change in any of the angles, nor in the magnifying power, and affects only the place where the images are formed. And, on the other hand, a lens placed at f, or F', where a real image is formed, makes no change in the places of the images, but affects the mutual inclination of the pencils. This affords a resource to the artist, by which he may combine properties which seem incompatible. The aperture of A determines the visible field and all the other apertures.

We must avoid forming a real image, such as fg, or F'G', on or very near any glass; for we cannot see this image without seeing along with it every particle of dust and every scratch on the glass. We see them as making part of the object when the image is exactly on the glass, and we see them confusedly, and so as to confuse the object, when the image is near it. For when the image is on or very near any glass, the pencil of light occupies a very small part of its surface, and a particle of dust intercepts a great proportion of it.

It is plain that this construction will not do for the telescope of graduated instruments, because the micrometer cannot be applied to the second image fg, on account of its being a little distorted, as has been observed of the Huyghenian eye-piece. Also the interposition of the glass C makes it difficult to correct the dispersion.

By proper reasoning from the correction in the Huyghenian eye-piece, we are led to the best construction of one with three glasses, which we shall now consider, taking it in a particular form, which shall make the discussion easy, and make us fully masters of the principles which lead to a better form. Therefore let PA (fig. 25) be the glass which first receives the light proceeding from the image formed by the object-glass, and let OP be the axis of the extreme pencil. This is refracted into PR, which is again refracted into Rr by the next lens Br. Let b be the focus of parallel rays of the second lens. Draw PB.r. We know that Ab : bB = PB : Br, and that rays of one kind diverging from P will be collected at r. But if PR, PV be a red and a violet ray, the violet ray will be more refracted at V, and will cross the red ray in some intermediate point g of the line Rr. If therefore the first image had been formed precisely on the lens PA, we should have a second image at fg free from all coloured fringes.

If the refractions at P and R are equal (as in the common day-telescope), the dispersion at V must be equal to that at P, or the angle eVr = VPR. But we have ultimately RPV : RrV = BC : AB = (Bb : Ab by the focal theorem). Therefore gVr : grV (or gr : gV, or Cf : fB) = Bb : Ab, and AB : Ab = Rr : Rg.

This shows, by the way, the advantage of the common day-telescope. In this AB = 2Ab, and therefore f is the place of the last image which is free from coloured fringes. But this image will not be seen free from coloured fringes through the eye-glass Cr if f' be its focus: for had gr, gw Telescope been both red rays, they would have been parallel after refraction; but gw being a violet ray, will be more refracted. It will not indeed be so much deflected from parallelism as the violet ray, which naturally accompanies the red ray to r, because it falls nearer the centre. By computation its dispersion is diminished about 4th.

In order that gw may be made parallel to gr after refraction, the refraction at r must be such that the dispersion corresponding to it may be of a proper magnitude. How to determine this is the question. Let the dispersion at q be to the dispersion produced by the refraction at r (which is required for producing the intended magnifying power) as 1 to 9. Make 9 : 1 = ff' : f'C = f'C : CD, and draw the perpendicular Dr' meeting the refracted ray rr' in r'. Then we know by the common focal theorem, that if f' be the focus of the lens Cr, red rays diverging from g will be united in r'. But the violet ray gw will be refracted into ev' parallel to rr'. For the angle ev'r : evr = (ultimately) f'C : CD = 9 : 1. Therefore the angle evr is equal to the dispersion produced at r, and therefore equal to r'ev', and ev' is parallel to rr'.

But by this we have destroyed the distinct vision of the image formed at fg, because it is no longer at the focus of the eye-glass. But distinct vision will be restored by pushing the glasses nearer to the object-glass. This makes the rays of each particular pencil more divergent after refraction through A, but scarcely makes any change in the directions of the pencils themselves. Thus the image comes to the focus f', and makes no sensible change in the dispersions.

In the common day-telescope, the first image is formed in the anterior focus of the first eye-glass, and the second image is at the anterior focus of the last eye-glass. If we change this last for one of half the focal distance, and push in the eye-piece till the image formed by the object-glass is half way between the first eye-glass and its focus, the last image will be formed at the focus of the new eye-glass, and the eye-piece will be achromatic. This is easily seen by making the usual computations by the focal theorem. But the visible field is diminished, because we cannot give the same aperture as before to the new eye-glass; but we can substitute for it two eye-glasses like the former, placed close together. This will have the same focal distance with the new one, and will allow the same aperture that we had before.

On these principles may be demonstrated the correction of colour in eye-pieces with three glasses of the following construction.

Let the glasses A and B be placed so that the posterior focus of the first nearly coincides with the anterior focus of the second, or rather so that the anterior focus of B may be at the place where the image of the object-glass is formed, by which situation the aperture necessary for transmitting the whole light will be the smallest possible. Place the third, C, at a distance from the second, which exceeds the sum of their focal distances by a space which is a third proportional to the distance of the first and second, and the focal distance of the second. The distance of the first eye-glass from the object-glass must be equal to the product of the focal distance of the first and second divided by their sum.

Let Oa, Aa, Bb, Cc, the focal distances of the glasses, be O, a, b, c. Then make AB = a + b nearly; BC = b + c + b + c; OA = be. The amplification or magnifying power will be = ob, the equivalent eye-glass = ac, and the field of vision = 3438 × aperture of A loc. dist. ob. gl. These eye-pieces will admit the use of a micrometer at the place of the first image, because it has no distortion.

Mr Dollond was anxious to combine this achromatism of the eye-pieces with the advantages which he had found in the eye-pieces with five glasses. The eye-piece of three glasses necessarily has a very great refraction at the glass B, where the pencil which has come from the other side of the axis must be rendered again convergent, or at least parallel to it. This occasions considerable aberrations, which may be avoided by giving part of this refraction to a glass put between the first and second, in the same way as was done by the glass B put between A and C in his five-glass eye-piece. But this deranges the whole process. His ingenuity, however, surmounted the difficulty, and he made eye-pieces of four glasses, which seem as perfect as can be desired. He did not publish his ingenious investigation; we imagine therefore that it will be an acceptable thing to the artists to have precise instructions how to proceed.

It is evident, that if we make the rays of different colours unite on the surface of the last eye-glass but one, commonly called the field-glass, the thing will be done, because the dispersion from this point of union will then unite with the dispersion produced by this glass alone; and this increased dispersion may be corrected by the last eye-glass in the way already shown.

Therefore let A, B (fig. 26) be the stations which we have fixed on for the first and second eye-glasses, in order to give a proper portion of the whole refraction to the second glass. Let b be the anterior focus of B. Draw PB through the centre of B. Make Ab : bB = AB : BK. Draw the perpendicular Kr, meeting the refracted ray in r. We know by the focal theorem, that red rays diverging from P will converge to r; but the violet ray PV, being more refracted, will cross Rr in some point g. Drawing the perpendicular fg, we get f for the proper place of the field-glass. Let the refracted ray Rr, produced backward, meet the ray OP coming from the centre of the object-glass in O. Let the angle of dispersion RPV be called p, and the angle of dispersion at V, that is, rVv, be v, and the angle VrR be r.

It is evident that OR : OP = p : v, because the dispersions are proportional to the sines of the refractions, which in this case are very nearly as the refractions themselves.

Let $\frac{OP}{OR} \left( \text{or} \frac{op}{pB} \text{or} \frac{pB}{bB} \right)$ be made = m. Then $v = mp$;

also $p : r = BK : AB = bB : Ab$, and $r = p \frac{Ab}{bB}$, or making $\frac{Ab}{bB} = n$, $r = np$; therefore $v : r = m : n = \frac{pB}{bB} : \frac{Ab}{bB} = pB : Ab$.

The angle RgV = gVr + grV = p(m + n); and RgV : Rrv = Rr : Rg, or $m + n : n = Rr : Rg$, and $Rg = Rr \frac{n}{m + n}$. But Rr is ultimately = BK = AB $\frac{bB}{Ab} = \frac{AB}{m + n}$,

therefore $Rg = \frac{AB}{n} \cdot \frac{n}{m + n} = \frac{n}{m + n}$, and $Bf = \frac{AB}{m + n}$.

This value of Bf is evidently = bB $\frac{AB}{pB + Ab}$. Now bB being a constant quantity while the glass B is the same, the place of union varies with $\frac{AB}{pB + Ab}$. If we remove B a little farther from A, we increase AB, and pB, and Ab, each by the same quantity. This evidently diminishes Bf. On the other hand, bringing B nearer to A increases Bf. If we keep the distance between the glasses the same, but increase the focal distance bB, we augment Bf, because this change augments the numerator and diminishes the denominator of the fraction $\frac{2B \cdot AB}{pB + Ab}$.

In this manner we can unite the colours at what distance we please, and consequently can unite them in the place of the intended field-glass, from which they will diverge with an increased dispersion, viz. with the dispersion competent to the refraction produced there, and the dispersion $p(m + n)$ conjoined.

It only remains to determine the proper focal distances of the field-glass and eye-glass, and the place of the eyeglass, so that this dispersion may be finally corrected.

This is an indeterminate problem, admitting of an infinity of solutions. We shall limit it by an equal division of the two remaining refractions, which are necessary in order to produce the intended magnifying power. This construction has the advantage of diminishing the aberration. Thus we know the two refractions, and the dispersion competent to each; it being nearly $\frac{1}{4}$th of the refraction. Call this q. The whole dispersion at the field-glass consists of q, and of the angle KgV of fig. 19, which we also know to be $= p(m + n)$. Call their sum s.

Let fig. 27 represent this addition to the eye-piece. Cg

Fig. 27.

is the field-glass coming in the place of fg of fig. 26, and Rgve is the red ray coming from the glass BR. Draw gs parallel to the intended emergent pencil from the eye-glass; that is, making the angle Csg with the axis correspond to the intended magnifying power. Bisect this angle by the line gK. Make sg : gg = s : q, and draw gK, cutting Cg in t. Draw tdD, cutting gk in d, and the axis in D. Draw dd' and Dr perpendicular to the axis. Then a lens placed in D, having the focal distance Dd', will destroy the dispersion at the lens gc, which refracts the ray gw into gr.

Let ge be the violet ray, making the angle grv = v. It is plain, by the common optical theorem, that gr will be refracted into rr' parallel to dd'. Draw gDr' meeting rr', and join rr'. By the focal theorem, two red rays gr, gw will be united in rr'. But the violet ray gw will be more refracted, and will take the path rr', making the angle of dispersion rr'v = v very nearly, because the dispersion at v does not sensibly differ from that at r. Now, in the small angles of refraction which obtain in optical instruments, the angles rr'e, rge are very nearly as gr and rr', or as gD and Dr', or as CD and DT; which, by the focal theorem, are as Cd and dd'; that is, $Dd : dc = rge : rr'$; But $Dd : dc = Ds : ds = sg : gg = s : q$. But rge = s; therefore rr'v = q = rr'v'c, and rr'v' is parallel to rr', and the whole dispersion at g is corrected by the lens Dr. The focal distance Cc of Cg is had by drawing Cx parallel to Kg, meeting Rg in x, and drawing xe perpendicular to the axis. It is easy to see that this (not inelegant) construction is not limited to the equality of the refractions $tegr$, $Krr'$. In whatever proportion the whole refraction $wgs$ is divided, we always can tell the proportion of the dispersions which the two refractions occasion at $g$ and $r$, and can therefore find the values of $s$ and $q$. Indeed this solution includes the problem in p. 167, col. 2, par. 2; but it had not occurred to us till the present occasion. Our readers will not be displeased with this variety of resource.

The intelligent reader will see that in this solution some quantities and ratios are assumed as equal which are not strictly so, in the same manner as in all the elementary optical theorems. The parallelism, however, of $wv'$ and $rr'$ may be made accurate by pushing the lens $Dr$ nearer to $Cg$, or retiring it from it. We may also, by pushing it still nearer, induce a small divergency of the violet ray, so as to produce accurate vision in the eye, and may thus make the vision through a telescope more perfect than with the naked eye, where dispersion is by no means avoided. It would therefore be an improvement to have the eye-glass in a sliding tube for adjustment. Bring the telescope to distinct vision; and if any colour be visible about the edges of the field, shift the eye-glass till this colour is removed. The vision may now become indistinct; but this is corrected by shifting the place of the whole eye-piece.

We have examined trigonometrically the progress of a red and a violet ray through many eye-pieces of Dollond's and Ramsden's best telescopes, and we have found in all of them that the colours are united on or very near the field-glass; so that we presume that a theory somewhat analogous to ours has directed the ingenious inventors. We meet with many made by other artists, and even some of theirs, where a considerable degree of colour remains, sometimes in the natural order, and often in the contrary order. This must happen in the hands of mere imitators, ignorant of principle. We presume that we have now made this principle sufficiently plain.

Fig. 28 represents the eye-piece of a very fine spy-glass by Mr Ramsden; the focal length of its object-glass is $\frac{3}{4}$ inches, with $\frac{1}{3}$th of aperture, $2^\circ 05'$ of visible field, and $15\cdot4$ magnifying power. The distances and focal lengths are of their proper dimensions, but the apertures are larger, that the progress of a lateral pencil might be more distinctly drawn. The dimensions are as follow:

Focal lengths $Aa = 0\cdot775$, $Bb = 1\cdot025$, $Cc = 1\cdot01$, $Dd = 0\cdot79$.

Distances $AB = 1\cdot18$, $BC = 1\cdot83$, $CD = 1\cdot105$.

It is perfectly achromatic, and the colours are united, not precisely at the lens $Cg$, but about $\frac{1}{3}$th of an inch nearer the eye-glass.

It is obvious that this combination of glasses may be used as a microscope; for if, instead of the image formed by the object-glass at $FG$, we substitute a small object, illuminated from behind, as in compound microscopes, and if we draw the eye-piece a very small way from this object, the pencils of parallel rays emergent from the eye-glass $D$ will become convergent to very distant points, and will there form an inverted and enlarged picture of the object, which may be viewed by a Huyghenian eye-piece; and we may thus get high magnifying powers without using very deep glasses.

We tried the eye-piece of which we have given the dimensions in this way, and found that it might be made to magnify 180 times with very great distinctness. When used as the magnifier of a solar microscope, it infinitely surpasses every thing we have ever seen. The picture formed by a solar microscope is generally so indistinct, that it is fit only for amusing ladies; but with this magnifier it seemed perfectly sharp. We therefore recommend this to the artists as a valuable article of their trade.

The only thing which remains to be considered in the theory of refracting telescopes, is the forms of the different lenses. Hitherto we have had no occasion to consider any thing but their focal distances; but their aberrations depend greatly on the adjustment of their forms to their situations. When the conjugate focuses of a lens are determined by the services which it is to perform, there is a certain form or proportion between the curvatures of their anterior and posterior surfaces, which will make their aberrations the smallest possible.

It is evident that this proportion is to be obtained by making the fluxion of the quantity within the parenthesis in the formula at the top of col. 2, p. 151, equal to nothing. When this is done, we obtain this formula for $a$, the radius of curvature for the anterior surface of a lens:

$$\frac{1}{a} = \frac{2m^2 + m}{2m + 4} + \frac{4m + 4}{2(m + 4)r},$$

where $m$ is the ratio of the sine of incidence to the sine of refraction, and $r$ is the distance of the focus of incident rays, positive or negative, according as they converge or diverge, all measured on a scale of which the unit is $n$, half of the radius of the equivalent isosceles lens.

It will be sufficiently exact for our purpose to suppose $m = \frac{3}{2}$ though it is more nearly $\frac{31}{20}$. In this case,

$$\frac{1}{a} = \frac{6}{7} + \frac{10}{7r} = \frac{6r + 10}{7r}.$$ Therefore $a = \frac{7r}{6r + 10}$, and $\frac{1}{b} = \frac{1}{a} - 1 = \frac{1 - a}{a}$.

As an example, let it be required to give the radii of curvature in inches for the eye-glass $be$ of page 164, col. 1, par. 2, which we shall suppose of $1\frac{1}{2}$ inch focal distance, and that $ee (= r)$ is $3\frac{1}{2}$ inches.

The radius of curvature for the equivalent isosceles lens is $1\cdot5$, and its half is $0\cdot75$. Therefore $r = \frac{31}{0\cdot75} = 5$; and our formula is $a = \frac{7 \times 5}{6 \times 5 + 10} = \frac{35}{40} = 0\cdot875$; and

$$\frac{1}{b} = \frac{1 - a}{a} = \frac{0\cdot125}{0\cdot875} = \frac{0\cdot875}{0\cdot125} = 7.$$ These values are parts of a scale of which the unit is $0\cdot75$ inch. Therefore

$a$, in inches, $= 0\cdot875 \times 0\cdot75 = 0\cdot65625,$

$b$, in inches, $= 7 \times 0\cdot75 = 5\cdot25.$

And here we must observe that the posterior surface is concave; for $b$ is a positive quantity, because $1 - a$ is a positive quantity as well as $a$; therefore the centre of sphericity of both surfaces lies beyond the lens.

And this determination is not very different from the usual practice, which commonly makes this lens a plane convex with its flat side next the eye; and there will not be much difference in the performance of these two lenses; for in all cases of maxima and minima, even a pretty considerable change of the best dimensions does not make a sensible change in the result. The same consideration leads to a rule which is very simple, and sufficiently exact for ordinary situations. This is to make the curvatures such, that the incident and emergent pencils may be nearly equally inclined to the surfaces of the lens. Thus in the eye-piece with five glasses, A and B should be most convex on their anterior sides; C should be most convex on the posterior side; D should be nearly isosceles; and E nearly plano-convex.

But this is not so easy a matter as appears at first sight. The lenses of an eye-piece have not only to bend the several pencils of light to and from the axis of the telescope; they have also to form images on the axes of these pencils. These offices frequently require opposite forms, as mentioned in par. 3, col. 2, p. 162. Thus the glass A, fig. 23, should be most convex on the side next the object, that it may produce little distortion of the pencils. But it should be most convex next the eye, that it may produce distinct vision of the image FG, which is very near it. This image should have its concavity turned towards A, whereas it is towards the object-glass. We must therefore endeavour to make the vertical image firmer, or even convex. This requires a glass very flat before and convex behind. For similar reasons the object-glass of a microscope and the simple eye-glass of an astronomical telescope should be formed in the same way.

This is a subject of most difficult discussion, and requires a theory for which our limits do not afford room. The artists are obliged to grope their way. The proper method of experiment would be, to make eye-pieces of large dimensions, with extravagant apertures to increase the aberrations, and to provide for each station A, B, C, and D, a number of lenses of the same focal distance, but of different forms; and we would advise making the trial in the way of a solar microscope; and to have two eye-pieces on trial at once. Their pictures can be formed on the same screen, and accurately compared; whereas it is difficult to keep in remembrance the performance of one eye-piece, and compare it with another.

We cannot add any thing to what Dr Smith has delivered on the theory of reflecting telescopes. There appears to be the same possibility of correcting the aberration of the great speculum by the contrary aberration of a convex small speculum, that we have practised in the compound object-glass of an achromatic refracting telescope. But this cannot be, unless we make the radius of the convex speculum exceedingly large, which destroys the magnifying power and the brightness. This therefore must be given up. Indeed their performance, when well executed, does already surpass all imagination. Dr Herschel found great advantages in what he called the front view, not using a plane mirror to throw the pencils to one side. But this cannot be practised in any but telescopes so large that the loss of light occasioned by the interposition of the observer's head may be disregarded.

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