ATTRACTION

ATTRACTION OF SPHERES.

Attraction of Spheres. 11. Spheres of the same homogeneous matter, attract particles placed on their surfaces, with forces proportional to their radii.

Spheres being similar solid figures, this proposition is no more than a particular case of what was before proved (8.).

Cor. If the spheres have different densities, the attractions at their surfaces are proportional to their radii multiplied by their densities (8. Cor.).

12. The force with which a particle, placed anywhere within a sphere of homogeneous matter, is urged towards the centre, is proportional to its distance from the centre.

Conceive a concentric sphere to be described, which contains the attracted particle in its surface; the matter between the two surfaces will exert no force on the particle (9.), which will therefore be urged to the centre, only by the attraction of the inner sphere, in the surface of which it is placed: but this force is proportional to the radius of the sphere, or to the distance of the particle from the centre (11.).

13. Let PNQ and ABC (fig. 7.) be two spheres of the same homogeneous matter, which attracts in the inverse proportion of the square of the distance: let the centres of the spheres be at M and D, and take MI equal to the radius of the sphere ABC, and ED equal to the radius of the sphere PNQ; the

attractions of the spheres upon particles placed at R and E are to one another as the squares of the radii of the spheres.

In the spheres draw two great circles perpendicular to the diameters PQ and AC, that pass through the points R and E; and let PpQ and AbC be two great circles, making equal indefinitely small angles NMP and BDd with the great circles PNQ and ABC. Let HK and FG, parallel to PQ and AC, be any two chords of the circles PpQ and AbC, that subtend similar arcs, or arcs containing the same number of degrees; and through HK and FG, let planes perpendicular to the circles PpQ and AbC, be drawn to cut the portions of the spheres contained in the angles NMP and BDd: join RH, RK, MH, MK, DF, DG, EF, EG. Because the arcs subtended by HK and FG, are like parts of their circumferences, it is plain, that the angle RMH = EDF, and RMK = EDG. And because ED = MH = MK, and RM = DF = DG (hyp.), therefore RH = EF, and RK = EG.

Conceive the chords HK and FG, together with the planes passing through them, to change their place a little, so as to describe two slender prisms, or elements of the portions of the spheres contained in the angles NMP and BDd. It is plain that MR and DO, the distances of the chords HG and FG from the centres of their circles, are constantly proportional to MN and DB, the radii of the spheres; wherefore RT and OS, the perpendicular sections of the small prisms, are similar figures, and have to one another the same ratio that $MN^2$ has to $DO^2$, or $MN^2$ to $DB^2$. Now, the attraction of the prism HK urging a particle at R to the centre M, is $=$

$XT\; \backslash times\; \backslash left\backslash \{\; \backslash frac\{1\}\{RH\}\; -\; \backslash frac\{1\}\{RK\}\; \backslash right\backslash \}$ (10.); and the attraction of

the prism FG urging a particle at E to the centre D,

is $=\; OS\; \backslash times\; \backslash left\backslash \{\; \backslash frac\{1\}\{EF\}\; -\; \backslash frac\{1\}\{EG\}\; \backslash right\backslash \}$. But, in consequence

of what was proved, $\backslash frac\{1\}\{RH\}\; -\; \backslash frac\{1\}\{RK\}\; =\; \backslash frac\{1\}\{EF\}\; -\; \backslash frac\{1\}\{EG\}$;

wherefore the attractions of the prisms are to one another as XT to OS, or as $MN^2$ to $DB^2$.—The same thing may be proved of all the elements of the two portions of the spheres contained in the angles NMP and BDd; wherefore those portions attract particles at R and E with forces proportional to the squares of the radii of the spheres. But because the small angles NMP and BDd are equal, each of the spheres may be divided into an equal number of such portions; wherefore the attractions of the whole spheres upon particles placed at R and E, are proportional to the squares of the radii of the spheres.

Cor. This proposition is true, when the particles of matter attract one another with forces proportional to any proposed function of the distance.

Let $RH\; =\; EF\; =\; f$, and $RK\; =\; EG\; =\; f\text{'}$; then, adopting the same notation as before (8. Cor.), the attractions of the prisms urging particles placed at R and E to the centres M and D, are respectively $XT\; \backslash times\; \backslash pm\; \backslash left\backslash \{\; \backslash Psi(f)\; -\; \backslash Psi(f\text{'})\; \backslash right\backslash \}$, and $OS\; \backslash times\; \backslash pm\; \backslash left\backslash \{\; \backslash Psi(f)\; -\; \backslash Psi(f\text{'})\; \backslash right\backslash \}$; consequently, those attractions have the same pro-

Attraction. portion that RT has to SO, or MN² to DB². Wherefore the attractions of the whole spheres are in the same proportion.

14. A particle placed anywhere without a sphere of homogeneous matter which attracts in the inverse proportion of the square of the distance, will be urged to the centre of the sphere, with a force that is inversely proportional to the square of the particle's distance from the centre.

Let ABC (Plate XXX. fig. 8.) be the sphere, O its centre, and P a particle without the sphere: conceive a concentric sphere PMN, of the same homogeneous matter with the sphere ABC, to be described with the radius PO. Then, by the last proposition, the attraction of the sphere ABC upon the particle P, is to the attraction of the sphere PMN upon a particle placed at A, as AO² to PO². But the attraction of the sphere PMN upon a particle placed at A, is equal to the attraction of the sphere ABC upon the same particle; for the attraction of the matter between the two spherical surfaces exerts no force upon a particle at A (9.). Wherefore, in the proportion set down above, the two middle terms are constantly the same wherever the point P is placed without the sphere ABC. Consequently, the first term of the proportion must follow the inverse ratio of the last term; that is, the attraction of the sphere ABC upon the external particle at P is inversely proportional to PO².

15. The same law of attraction being supposed, a homogeneous sphere will attract a particle placed without it, with the same force as if all the matter of the sphere were collected in the centre.

Let $f$ denote the distance of the particle from the centre; then it follows, from the last proposition, that the attraction of the sphere upon the particle will have, for its measure, $\backslash frac\{A\}\{f^2\}$; A denoting a constant quantity that will be determined by any particular case; that is, by the actual attractive force corresponding to any determinate distance from the centre. Let $r$ denote the radius of the sphere, and M its mass; then no part of the matter of the sphere being nearer the attracted particle than $(f-r)$, and none of it more remote than $(f+r)$, the attraction of the sphere on the particle will be

greater than $\backslash frac\{M\}\{(f+r)^2\}$, and less than $\backslash frac\{M\}\{(f-r)^2\}$. There-

fore $\backslash frac\{A\}\{f^2\}$ is always contained between those limits,

which requires that $A\; =\; M$. For, if A were greater than M, such values of $f$ might be found as would

make $\backslash frac\{A\}\{f^2\}$ equal to, or greater than $\backslash frac\{M\}\{(f-r)^2\}$; and, if A were less than M, such values of $f$ might be found

as would make $\backslash frac\{A\}\{f^2\}$ equal to, or less than $\backslash frac\{M\}\{(f+r)^2\}$. Therefore $A\; =\; M$; and the attraction of the sphere

is equal to $\backslash frac\{M\}\{f^2\}$, or the same as if all the matter were

collected in the centre.

VOL. I. PART II.

If the radius of the sphere = $r$, the density of the matter contained in it = $d$; then the mass, or $M\; =$

$\backslash frac\{4\backslash pi\; r^3\; d\}\{3\}$ ($\backslash pi$ being the circumference of the circle whose diameter is unit), and the attraction of the sphere at

the distance $f$ from the centre = $\backslash frac\{4\backslash pi\; r^3\; d\}\{3f^2\}$. This is still

true at the surface of the sphere when $f\; =\; r$, so that

the attraction at the surface = $\backslash frac\{4\backslash pi\; r\; d\}\{3\}$; which expres-

sion, with the help of what is proved in (12.), enables us to compare the intensities of the attractions of homogeneous spheres, at all distances from the centre, without or within the surfaces.

Cor. 1. A shell of homogeneous matter contained between two concentric spherical surfaces, will attract a particle placed without it, with the same force as if all the matter of the shell were collected in its centre.

For the attractive force of such a shell is equal to the difference of the attractions of two concentric spheres of the same homogeneous matter with the shell.

Cor. 2. A sphere composed of concentric shells, that vary in their densities according to any law, will attract a particle placed without it, with the same force as if all the matter were collected in the centre.

For this having been proved of one shell (Cor. 1.), it must be true of any number of shells.

If $\backslash phi(r)$ denote the density at the distance $r$ from the centre, the quantity of matter in the sphere will be $4\backslash pi\; \backslash int\; \backslash phi(r)\; \backslash cdot\; r^2\; dr$; and the attraction on a particle without the sphere at the distance $f$ from the

centre = $\backslash frac\{4\backslash pi\; \backslash int\; \backslash phi(r)\; \backslash cdot\; r^2\; dr\}\{f^2\}$.

16. Two spheres, each composed of concentric shells of variable density, attract one another with the same force as if all the matter of each were collected in its centre.

For the attraction of a sphere A upon every particle of another sphere B will remain the same, if we suppose all the matter of A to be collected in its centre (15.). But the attraction of any particles of matter placed in A's centre, upon the sphere B, is equal and opposite to the attraction of B, upon the same matter so placed: and, again, the attraction of B upon all the particles placed in the centre of A, will remain unchanged, if we suppose the matter of B to be collected in its centre. Wherefore A attracts B with the same force as if the matter of each were collected in its centre.

17. Supposing that the particles of matter attract with a force proportional to the distance, a body of any shape will attract a particle of matter placed anywhere with the same force, and in the same direction, as if all the matter of the body were collected in its centre of gravity.

Suppose that the attracted particle is placed at P (fig. 9.), and the centre of gravity of the attracting body at G: join PG, and let any plane pass through that line. Let I be a small part, or element of the body, and

Attraction. from L draw $LK$ perpendicular to the plane passing through $PG$, and $KF$ perpendicular to $PG$; join $PL$ and $PK$. Put $dm$ to denote the quantity of matter, or the mass of the element $L$; then its attractive force, urging the particle in the direction $PL$, is $=\; PL\; \backslash times\; dm$, which, by the resolution of forces, is equivalent to the two forces, $PK\; \backslash times\; dm$ and $KL\; \backslash times\; dm$; and, again, the single force $PK\; \backslash times\; dm$ is equivalent to the two forces $FK\; \backslash times\; dm$, and $PF\; \backslash times\; dm\; =\; PG\; \backslash times\; dm\; +\; GF\; \backslash times\; dm$. Therefore, the attraction of the element $L$, upon the particle at $P$, is equivalent to these four separate forces, viz. $PG\; \backslash times\; dm$, $GF\; \backslash times\; dm$, $FK\; \backslash times\; dm$, $KL\; \backslash times\; dm$, which urge the particle $P$ respectively in the directions, $PG$, $GF$, $FK$, $KL$. But, from the nature of the centre of gravity, the sum of all the forces, $KL\; \backslash times\; dm$, that urge the particle $P$ to one side of the plane passing through $PG$, is just equal to the sum of the forces that urge it to the other side of the same plane; and the sum of all the forces, $FK\; \backslash times\; dm$, that urge $P$ to one side of the line $PG$, is just equal to the sum of the forces that urge it to the other side of the same line; and the sum of all the forces, $GF\; \backslash times\; dm$, that urge $P$ towards the point $G$, is just equal to the sum of the forces that urge it from the same point. Wherefore all the preceding forces mutually destroy one another, excepting the forces, $PG\; \backslash times\; dm$, the sum of which, when extended to all the elements of the attracting body, is $=\; PG\; \backslash times$ mass of the body. Wherefore the whole attraction upon $P$ is the same as if all the matter of the body were collected in its centre of gravity.

Cor. Supposing that the particles of matter attract with a force proportional to the distance, a homogeneous sphere will attract a particle placed anywhere in the same manner as if all the matter of the sphere were collected in the centre.

For the centre of gravity of a homogeneous sphere, is the same as the centre of its figure. This corollary is likewise true of a sphere composed of concentric shells of variable density; and it is easy to apply the demonstration of (16.) to prove that, in this law of attraction, two spheres, each composed of concentric shells of variable density, will attract one another with the same force as if the matter of each were collected in its centre.

18. To investigate what are the laws of attraction, in regard to the distance, according to which a shell of homogeneous matter, contained between two concentric spherical surfaces, will attract a particle placed without it, in the same manner as if all the matter of the shell were collected in the centre.

It has been proved that this property actually belongs to homogeneous shells in the law of attraction which obtains in nature, and likewise when the particles of matter attract with a force proportional to the distance; but it is interesting to know whether it is confined to these two cases alone, or extends to other laws of attraction. This can only be discovered by a direct analysis.

Let $r\; =\; PC$ (fig. 10.), the distance of the attracted point from the centre of the shell; $u\; =\; CA$ the radius of the inner surface of the shell; $f\; =\; PM$, the distance of $P$ from any point in the surface. Having drawn the diameter $AD$ through $P$, let $AMD$ and $AND$ be two great circles, making, with one another,

an indefinitely small angle $MAN\; =\; dq$; and let two small circles $BMG$, $bmG$, indefinitely near one another, of which $A$ and $D$ are the poles, meet the former circles in $M$, $N$, $m$, $n$; and draw $MS$, $NS$ to the centre of the circle $BMG$. Put $\backslash delta$ for the measure of the arc $AM$; then $MS\; =\; u\; \backslash sin.\; \backslash delta$; $MN\; =\; dq\; \backslash cdot\; u\; \backslash sin.\; \backslash delta$; $Mm\; =\; u\; d\backslash delta$; and the quadrilateral space $MNmm\; =\; u^2\; \backslash cdot\; dq\; \backslash cdot\; d\backslash delta\; \backslash sin.\; \backslash delta$. We may suppose the thickness of the shell indefinitely small; since, if the property belong to an elementary shell indefinitely thin, it will be true of one of a determinate thickness, which can be resolved into such elements. Suppose the thickness of the shell to be $=\; du$; then, the quantity of matter in the part standing upon the quadrilateral space $MNmm\; =\; u^2\; du\; \backslash cdot\; dq\; \backslash cdot\; d\backslash delta\; \backslash sin.\; \backslash delta$. Let $\backslash phi(f)$ represent the direct attraction of a particle at $M$ in the direction $PM$; then its attraction directed to the

$$\backslash text\{centre\; \}\; C\; =\; \backslash phi(f)\; \backslash times\; \backslash frac\{PS\}\{PM\}\; =\; \backslash frac\{r\; -\; u\; \backslash cos.\; \backslash delta\}\{f\}\; \backslash times\; \backslash phi(f);$$

and the attraction of the element of the shell in the same direction $=\; u^2\; du\; \backslash cdot\; dq\; \backslash cdot\; d\backslash delta\; \backslash sin.\; \backslash delta\; \backslash times\; \backslash frac\{r\; -\; u\; \backslash cos.\; \backslash delta\}\{f\}\; \backslash cdot\; \backslash phi(f)$.

This expression is proportional to $dq$, when $\backslash delta$ and $f$ remain constant; and, therefore (denoting by $\backslash sigma$ the circumference of the circle whose diameter is unit), the attraction of the whole zone contained between the small circles $BMG$, $bmG$, will be $=\; 2\backslash pi\; \backslash cdot\; u^2\; du\; \backslash cdot\; d\backslash delta\; \backslash sin.\; \backslash delta$.

$\backslash frac\{r\; -\; u\; \backslash cos.\; \backslash delta\}\{f\}\; \backslash cdot\; \backslash phi(f)$; and the attractive force of the whole shell will be

$$2\backslash pi\; \backslash cdot\; u^2\; du\; \backslash int\; d\backslash delta\; \backslash sin.\; \backslash delta\; \backslash cdot\; \backslash frac\{r\; -\; u\; \backslash cos.\; \backslash delta\}\{f\}\; \backslash cdot\; \backslash phi(f);$$

the fluent to be extended from $\backslash delta\; =\; 0$ to $\backslash delta\; =\; \backslash pi$.

Again, the quantity of matter in the shell is $=\; 4\backslash pi\; \backslash cdot\; u^2\; du$; and the attraction of this matter placed in the centre, at the distance $r$ from $P$, is $=\; 4\backslash pi\; \backslash cdot\; u^2\; du\; \backslash cdot\; \backslash phi(r)$.

If now we equate the attraction of the shell, to the attraction of its matter placed in the centre, and leave out the factors common to both, we shall get

$$2\backslash pi(r)\; =\; \backslash int\; d\backslash delta\; \backslash sin.\; \backslash delta\; \backslash cdot\; \backslash frac\{r\; -\; u\; \backslash cos.\; \backslash delta\}\{f\}\; \backslash cdot\; \backslash phi(f)$$

the limits of the integral being the same as before.

But $f^2\; =\; r^2\; -\; 2ru\; \backslash cos.\; \backslash delta\; +\; u^2$; then $d\backslash delta\; \backslash sin.\; \backslash delta\; =\; \backslash frac\{df\}\{r\}$; also $r\; -\; u\; \backslash cos.\; \backslash delta\; =\; \backslash frac\{f^2\; +\; r^2\; -\; u^2\}\{2r\}$; wherefore,

by substitution, we get

$$4\; \backslash cdot\; r^2\; \backslash phi(r)\; \backslash cdot\; u\; =\; \backslash int\; (f^2\; +\; r^2\; -\; u^2)\; \backslash cdot\; df\; \backslash cdot\; \backslash phi(f);$$

or, which is equivalent, $4\; \backslash cdot\; r^2\; \backslash phi(r)\; \backslash cdot\; u\; =$

$$(f^2\; +\; r^2\; -\; u^2)\; \backslash cdot\; \backslash int\; df\; \backslash cdot\; \backslash phi(f)\; -\; 2\; \backslash int\; f\; df\; \backslash int\; df\; \backslash cdot\; \backslash phi(f),$$

the limits of this integral being from $f\; =\; r\; -\; u$ to $f\; =\; r\; +\; u$, which correspond to $\backslash delta\; =\; 0$ and $\backslash delta\; =\; \backslash pi$.

Now let $\backslash int\; df\; \backslash cdot\; \backslash phi(f)\; =\; \backslash Psi(f)$; and $\backslash int\; f\; df\; \backslash int\; df\; \backslash cdot\; \backslash phi(f)\; =\; \backslash int\; f\; df\; \backslash cdot\; \backslash Psi(f)\; =\; \backslash Psi\text{'}(f)$; then, by taking the fluents between the proper limits, we get

$$\backslash underbrace\{\backslash text\{Attraction.\}\}\_\{4r^2\backslash phi(r).u\; =\; 2r\; \backslash cdot\; \backslash left\backslash \{\; (r+u)\backslash Psi(r+u)\; -\; (r-u)\backslash Psi(r-u)\; \backslash right\backslash \}\; -\; 2\; \backslash left\backslash \{\; \backslash Psi\text{'}(r+u)\; -\; \backslash Psi\text{'}(r-u)\; \backslash right\backslash \}\; \}$$

If we develop the binomial functions in the last expression, all the even powers of $u$ will disappear, and the odd powers only will remain; these last terms being all contained in this general formula,

$$\backslash text\{viz.\; \}\; \backslash frac\{4\}\{1.2.3\; \backslash dots\; 2n+1\}\; \backslash cdot\; \backslash left\backslash \{\; r\; \backslash cdot\; \backslash frac\{d^\{2n+1\}\; r\; \backslash Psi(r)\}\{dr^\{2n+1\}\}\; -\; \backslash frac\{d^\{2n+1\}\; r\; \backslash Psi\text{'}(r)\}\{dr^\{2n+1\}\}\; \backslash right\backslash \}\; \backslash cdot\; u^\{2n+1\};$$

and, observing that $\backslash frac\{d\; \backslash cdot\; \backslash Psi\text{'}(r)\}\{dr\}\; =\; r\backslash Psi(r)$, the same expression will become $\backslash frac\{4\}\{1.2.3\; \backslash dots\; 2n+1\}\; \backslash cdot\; \backslash left\backslash \{\; r\; \backslash cdot\; \backslash frac\{d^\{2n+1\}\; r\; \backslash Psi(r)\}\{dr^\{2n+1\}\}\; -\; \backslash frac\{d^\{2n+1\}\; r\; \backslash Psi(r)\}\{dr^\{2n+1\}\}\; \backslash right\backslash \}\; \backslash cdot\; u^\{2n+1\};$

which, again, is more simply expressed thus, viz.

$$\backslash frac\{4r^2\}\{1.2.3\; \backslash dots\; 2n+1\}\; \backslash cdot\; \backslash frac\{d\}\{dr\}\; \backslash left\backslash \{\; \backslash frac\{1\}\{r\}\; \backslash cdot\; \backslash frac\{d^\{2n\}\; r\; \backslash Psi(r)\}\{dr^\{2n\}\}\; \backslash right\backslash \}\; \backslash cdot\; u^\{2n+1\}.$$

Wherefore, by substituting the development instead of the functions, and then, by dividing by $\backslash frac\{1\}\{4\}\; r^2\; u$, we get

From the nature of the function $\backslash Psi(r)$, we get $\backslash phi(r)\; =\; \backslash frac\{d\; \backslash cdot\; \backslash Psi(r)\}\{dr\}$; wherefore each of the remaining terms must be separately equal to nothing: Hence

$$\backslash frac\{d\}\{dr\}\; \backslash left\backslash \{\; \backslash frac\{1\}\{r\}\; \backslash cdot\; \backslash frac\{d^2\; r\; \backslash Psi(r)\}\{dr^2\}\; \backslash right\backslash \}\; =\; 0;$$

from which we find $r\; \backslash Psi(r)\; =\; \backslash frac\{1\}\{2\}\; A\; r^3\; +\; A\text{'}\; r\; -\; A\text{'}\text{'}$, $A,\; A\text{'},\; A\text{'}\text{'}$ being arbitrary constant quantities; and this value of $r\; \backslash Psi(r)$, it is plain, will likewise render all the succeeding terms of the development evanescent. Wherefore

$$\backslash phi(r)\; =\; \backslash frac\{d\; \backslash cdot\; \backslash Psi(r)\}\{dr\}\; =\; A\; r\; +\; \backslash frac\{A\text{'}\}\{r^2\}.$$

Thus the most general expression of the law of attraction, that possesses the property in question, is a combination of the two laws above mentioned, with each of which it coincides, according as we make the one or other of the constant quantities equal to nothing. We have therefore a direct proof, that the law of nature is the only one which will make the attraction decrease as the distance increases, and in which a spherical shell, or a sphere, will attract in the same

manner as if all the matter were collected in the Attraction centre.

Laplace has arrived at the same conclusion by a different process. (Mech. Celeste, Liv. 2d. Chap. 2. No. 12. Rem. Part.)

ATTRACTION OF SPHEROIDS OF REVOLUTION.

19. Let $APBQ$ (Plate XXXI. fig. 11.) and $CMDN$ Attraction be two concentric ellipses, similar to one another, and similarly situated, of which $AB$ and $CD$ are either the greater, or less, axes; and let $PQ$ be perpendicular to $AB$. Conceive the ellipses to revolve about $PQ$, so as to describe an indefinitely small angle; then, supposing the law of attraction to be inversely proportional to the square of the distance, the thin solid of homogeneous matter described by the ellipse $APBQ$, will attract a particle placed at $P$, in a direction perpendicular to any plane passing through $PQ$, with the same force that the thin solid of the same matter described by the ellipse $CMDN$, will attract a particle placed at $C$ perpendicularly to the same plane.

From $C$ draw $CM,\; CN$, making equal angles with $CD$, and $PR,\; PT$ respectively parallel to $CM,\; CN$; and let $Cm,\; Cn,\; Pr,\; Pt$, be drawn in the same manner, and indefinitely near the former lines. While the ellipses revolve about $PQ$, the small sectors will describe pyramids that have their vertices at $C$ and $P$. It is manifest that the pyramids so described are similar: for their angles at $C$ and $P$ in the planes of the ellipses are equal; and their other angles described by revolving about $PQ$ are likewise equal, because the sectors are equally inclined to that axis. Wherefore, the direct attractions of all the small pyramids upon the particles $P$ and $C$, are proportional to the lengths $PR,\; PT,\; CM,\; CN$ (7); and consequently the forces that urge the particles $P$ and $C$ in a direction at right angles to any plane passing through $PQ$, are proportional to the perpendiculars let fall upon that plane from $R,\; T,\; M,\; N$. But, because $PR,\; PT,\; CM,\; CN$ are equally inclined to $PQ$, they will make equal angles with any plane passing through $PQ$: wherefore the perpendiculars drawn to the plane from $R,\; T,\; M,\; N$, will be respectively proportional to $CM,\; CN,\; PR,\; PT$. But $CM\; +\; CN\; =\; PR\; +\; PT$ (6): wherefore, the sum of the perpendiculars drawn to the plane from $M$ and $N$, will be equal to the sum of the perpendiculars drawn to it from $R$ and $T$. Consequently the force of the pyramids $PR$ and $PT$, which urges the particle $P$ at right angles to the plane, is equal to the force of the pyramids $CM$ and $CN$, which urges the particle $C$ in a parallel direction. The same thing is true of all the small pyramids that make up the thin solids described by the ellipses $APBQ$ and $CMDN$; and it is therefore true of the whole solids.

It is to be observed, that when the pyramids $PR$ and $PT$ fall on opposite sides of $PQ$, it is the difference of their attractions which is equal to the sum of the attractions of $CM$ and $CN$; and it is the difference of the perpendiculars let fall from $T$ and $R$ on opposite sides of the plane, which is equal to the sum of the perpendiculars let fall from $M$ and $N$.

Attraction. 20. Let $APBQ$ be a spheroid of revolution, $PQ$ the axis of revolution, and $ACB$ a plane through the centre perpendicular to $PQ$ (Plate XXXI, fig. 12.). If $D$ be a particle in the surface of the spheroid, and $DL$ perpendicular to the plane $ACB$; then the attraction of the spheroid on a particle placed at the pole $P$, will be to the force with which a particle placed at $C$, is attracted in the direction $DL$, as $PC$ is to $DL$.

Through $D$ draw a plane parallel to the plane $ACB$, and let the plane so drawn cut the axis $PQ$ in $F$: draw the straight line $DFE$ to terminate in the spheroid, and describe another spheroid through $F$, having the same centre with the spheroid $APBQ$, and similar to it, and similarly situated. Conceive an indefinitely great number of planes, making indefinitely small angles with one another, to be drawn through $DE$, so as to divide the two spheroids into an indefinitely great number of thin solids, or slices: then the sections which every one of the planes make with the spheroids will be similar ellipses, having the same centre (5.): and it is manifest that a straight line drawn through $F$ at right angles to $DE$, in any one of the planes, will pass through the centre of the two ellipses contained in it, and will coincide with an axis of each. Wherefore, the force with which every one of the slices, or elements, of the spheroid $APBQ$ attracts a particle placed at $D$ in the direction $DL$, is equal to the force with which the corresponding slice, or element, of the spheroid $GFHK$, attracts a particle placed at $F$ in the direction $FC$ (19.). Wherefore, the whole attraction of the spheroid $APBQ$ upon a particle at $D$, in the direction $DL$, is equal to the whole attraction of the spheroid $GFHK$, upon a particle at $F$. But the attractions of the spheroids $APBQ$ and $GFHK$, upon particles placed at $P$ and $F$, are to one another as $PC$ to $FC$ (8.). Wherefore, the attraction of the spheroid $APBQ$ upon a particle at $P$, is to the force with which the same spheroid attracts a particle at $D$, in the direction $DL$, as $PC$ is to $FC$ or $DL$.

21. Let $APBQ$ be a spheroid of revolution, and $PQ$ the axis of revolution, as before. If $D$ be a particle in the surface, $ADPB$ (fig. 13.) a section through $D$, and the axis $PQ$, and $DL$ perpendicular to $PQ$; the attraction of the spheroid upon a particle at $A$, will be to the force with which a particle at $D$ is attracted, in the direction $DL$, as $AC$ is to $DL$.

Through $D$ draw a plane perpendicular to $AB$, which cuts the section $ADPB$ in the straight line $DFE$; and let a spheroid $GFHK$ be described through $F$, having the same centre with the spheroid $APBQ$, and similar to it, and similarly situated. Then, conceiving the two spheroids to be divided into an indefinitely great number of thin slices by planes passing through $DE$, the force with which every slice, or element, of the spheroid $APBQ$ attracts a particle at $D$ in the direction $DL$, will be equal to the force with which the corresponding slice, or element, of the spheroid $GFHK$ attracts a particle at $F$ (19.). But the attractions of the spheroids $APBQ$ and $GFHK$ upon particles placed at $A$ and $F$, are to one another as $AC$ to $CF$ (8.). Wherefore, the attraction of the spheroid $APBQ$ upon a particle at $A$, is to the force with which the same

spheroid attracts a particle at $D$, in the direction $DL$, as $AC$ to $FL$ or $DL$. Attraction.

The two last propositions will enable us to find both the direction and the intensity of the attraction of a homogeneous spheroid of revolution upon a particle placed anywhere on the surface, when we have ascertained the attractive forces at the poles, and at the circumference of the circular section made by a plane through the centre perpendicular to the axis. For the whole attraction at any point, is the compound force arising from the attractions perpendicular to the axis, and parallel to it. The next object of our research is, therefore, to determine the two forces above-mentioned, viz. the attraction at the poles, and at the circular section, equally distant from both poles.

22. Let $ABD$ be an indefinitely slender pyramid, of which the base $BD$ is perpendicular to the edge $AD$ (fig. 14.): let $B$ = base $BD$, and $f$ = length

$AD$; then $\backslash frac\{B\}\{f^2\}$ = the attraction of the whole matter of the pyramid upon a particle placed at the vertex $A$.

Let $AM\; =\; x$; then the section $MN$ parallel to the base $BD\; =\; \backslash frac\{B\; \backslash cdot\; x^2\}\{f^2\}$; and, $MP$ = element of the

prism = $\backslash frac\{B\; \backslash cdot\; x^2\; dx\}\{f^2\}$; and the attraction of the element

upon a particle placed at $A\; =\; \backslash frac\{MP\}\{AM^2\}\; =\; \backslash frac\{B\; \backslash cdot\; dx\}\{f^2\}$; the

fluent of which is $=\; \backslash frac\{Bx\}\{f^2\}\; =$ attraction of the pyramid

$AM$ upon a particle at $A$. And, when $x\; =\; f$, this

becomes $=\; \backslash frac\{B\}\{f\}\; =$ attraction of the pyramid $AD$ upon a particle placed at $A$.

23. To investigate the attraction of a homogeneous spheroid of revolution, upon a particle placed at the pole.

Let $P$ (fig. 15.) be the pole, $PCQ$ the axis of revolution, and $APBQ$ a section of the spheroid by a plane passing through $PQ$, and any point $M$, in the surface; draw $PM$, $Pm$ indefinitely near $PM$, and $Mm$ perpendicular to $Pm$. Conceive the plane $PMQ$ to revolve about $PQ$, so as to describe the indefinitely small angle $BCO$; then the small triangle $MPm$ will describe a slender pyramid, having its vertex at $P$, and of which the base is a rectangle, contained by $Mm$ and $RT$; for the point $M$ moving parallel to $R$, it will describe a line equal and parallel to that described by $R$, namely, to $RT$.

Let $PM\; =\; f$; and the angle $KPM$, which $PM$ makes with a perpendicular to the axis, $=\; \backslash theta$; and the indefinitely small angle $BCO\; =\; d\backslash phi$. Then $Mm\; =\; fd\backslash theta$; $RT\; =\; CR\; \backslash times\; d\backslash phi\; =\; f\; \backslash cos.\; \backslash theta\; \backslash cdot\; d\backslash phi$; and $B$, the base of the slender pyramid described by the triangle $MPm$, $=\; d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash cdot\; f^2$; wherefore, the direct attraction

of the pyramid on a particle at $P\; =\; \backslash frac\{B\}\{f\}\; (22)\; =$

Attraction, $d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash cdot\; f$; and the elementary attraction of the spheroid in the direction $PC\; =$ direct attraction of the pyramid $\backslash times\; \backslash frac\{PS\}\{PM\}\; =\; d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash theta\; \backslash cdot\; f$.

Again, let $MR\; =\; x$, $CR\; =\; y$, $PC\; =\; k$, $AC\; =\; k\text{'}$; then $y\; =\; f\; \backslash cos.\; \backslash theta$; $x\; =\; k\; -\; f\; \backslash sin.\; \backslash theta$; if we substitute these values in the equation of the solid (1.), we get $\backslash frac\{(k\; -\; f\; \backslash sin.\; \backslash theta)^2\}\{k^2\}\; +\; \backslash frac\{f^2\; \backslash cos.\; \backslash theta^2\}\{k\text{'}^2\}\; =\; 1$; whence

$$f\; =\; \backslash frac\{2k\text{'}^2\; k\}\{k^2\}\; \backslash cdot\; \backslash frac\{\backslash sin.\; \backslash theta\}\{\backslash cos.\; \backslash theta^2\; +\; \backslash frac\{k\text{'}^2\}\{k^2\}\; \backslash sin.\; \backslash theta^2\}.$$

By substituting the value of $f$ just found in the preceding expression of the elementary attraction of spheroid, it will become

$$\backslash frac\{2k\text{'}^2\; k\}\{k^2\}\; d\backslash phi\; \backslash cdot\; \backslash frac\{d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash cdot\; \backslash sin.\; \backslash theta\}\{\backslash cos.\; \backslash theta^2\; +\; \backslash frac\{k\text{'}^2\}\{k^2\}\; \backslash sin.\; \backslash theta^2\};$$

which must be integrated from $\backslash phi\; =\; 0$ to $\backslash phi\; =\; 2\backslash pi$; and from $\backslash theta\; =\; 0$ to $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{2\}$; $\backslash tau$ denoting always the half-circumference when radius is unit.

In an oblate spheroid $k$ is less than $k\text{'}$; put $k\text{'}^2\; -\; k^2\; =\; k^2\; \backslash cdot\; e^2$, and $z\; =\; \backslash sin.\; \backslash theta$; then the element of the attractive force will become, by substitution,

$$\backslash frac\{2k\text{'}^2\; k\}\{k^2\}\; d\backslash phi\; \backslash cdot\; \backslash frac\{z^2\; dz\}\{1\; +\; e^2\; z^2\}\; =\; \backslash frac\{2k\text{'}^2\; k\}\{k^2\; e^2\}\; d\backslash phi\; \backslash cdot\; \backslash left\backslash \{\; e\; dz\; -\; \backslash frac\{e\; dz\}\{1\; +\; e^2\; z^2\}\; \backslash right\backslash \};$$

and by integrating from $z\; =\; 0$ to $z\; =\; 1$, we get,

$$\backslash frac\{2k\text{'}^2\; k\}\{k^2\; e^2\}\; d\backslash phi\; \backslash cdot\; \backslash left\backslash \{\; e\; -\; \backslash text\{arc.\; tan.\; \}\; e\; \backslash right\backslash \};$$

for the force with which the matter between the planes $PBQ$ and $POQ$ urges the particle $P$ to the centre. Wherefore the whole attractive force of the spheroid upon a particle at $P$ is =

$$\backslash frac\{4\backslash pi\; \backslash cdot\; k\text{'}^2\; k\}\{k^2\; e^2\}\; \backslash cdot\; \backslash left\backslash \{\; e\; -\; \backslash text\{arc.\; tan.\; \}\; e\; \backslash right\backslash \};$$

And, because $\backslash frac\{4\backslash pi\; \backslash cdot\; k\text{'}^2\; k\}\{s\}\; =$ mass of the spheroid $=\; M$, we get the measure of the attraction of the oblate spheroid upon a particle placed at the pole, equal to

$$k\; \backslash cdot\; \backslash frac\{3M\}\{k^2\; e^2\}\; \backslash cdot\; \backslash left\backslash \{\; e\; -\; \backslash text\{arc.\; tan.\; \}\; e\; \backslash right\backslash \}.$$

In an oblong spheroid, $k$ is greater than $k\text{'}$; put $k^2\; -\; k\text{'}^2\; =\; k^2\; \backslash cdot\; e^2$; then the element of the attractive force will become, by substitution,

$$\backslash frac\{2k\text{'}^2\; k\}\{k^2\}\; d\backslash phi\; \backslash cdot\; \backslash frac\{z^2\; dz\}\{1\; -\; e^2\; z^2\}\; =\; \backslash frac\{2k\text{'}^2\; k\}\{k^2\; e^2\}\; d\backslash phi\; \backslash cdot\; \backslash left\backslash \{\; \backslash frac\{e\; dz\}\{1\; -\; e^2\; z^2\}\; -\; e\; dz\; \backslash right\backslash \};$$

whence, by proceeding as before, we get the measure of the attractive force of the oblong spheroid on a particle placed at the pole, equal to

$$k\; \backslash cdot\; \backslash frac\{3M\}\{k^2\; e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash frac\{1\}\{2\}\; \backslash text\{hyp.\; log.\; \}\; \backslash frac\{1+e\}\{1-e\}\; -\; e\; \backslash right\backslash \}.$$

Cor. In an oblate spheroid differing little from a sphere, $e^2$ will be a very small fraction, of which we

may reject the higher powers. When this is done, Attraction, the preceding expression of the polar attraction, viz.

$$\backslash frac\{4\backslash pi\; \backslash cdot\; k\; k\text{'}^2\}\{k^2\; e^2\}\; \backslash cdot\; (e\; -\; \backslash text\{arc.\; tan.\; \}\; e),$$

will be $=\; 4\backslash pi\; k\; \backslash cdot\; (1\; +\; e^2)\; \backslash left(\; \backslash frac\{1\}\{3\}\; -\; \backslash frac\{1\}\{3\}\; e^2\; \backslash right)\; =\; \backslash frac\{4\backslash pi\; k\}\{3\}\; \backslash cdot\; \backslash left(\; 1\; +\; \backslash frac\{2\}\{5\}\; e^2\; \backslash right)$.

And, if $k\text{'}\; =\; k\; +\; \backslash tau\; =\; k\; \backslash sqrt\{1\; +\; e^2\}$, be the radius of the equator, then $\backslash frac\{2\}\{k\}\; =\; e^2$; so that the attraction at the pole will be

$$\backslash frac\{4\backslash pi\; k\}\{3\}\; \backslash cdot\; \backslash left(\; 1\; +\; \backslash frac\{4\}\{5\}\; \backslash cdot\; \backslash frac\{\backslash tau\}\{k\}\; \backslash right).$$

24. To investigate the attraction of a homogeneous spheroid of revolution, on a particle placed in the circumference of the circular section, made by a plane through the centre, at right angles, to the axis of revolution.

Let $P$ (Plate XXXI. fig. 16.) be the pole, $PC$ the axis of revolution. $A$, a point in the circular section $AOB$, made by a plane through the centre perpendicular to $PC$. Let $M$ be any point in the surface of the spheroid: $AMO$ a section through $A$ and $M$ by a plane perpendicular to $AOB$; $Am$ a line in that plane indefinitely near $AM$, and $Mm$ perpendicular to $Am$; $MR$ perpendicular to $AO$, and $RS$ to $AB$. Conceive the plane $AMO$ to revolve about $A$, so as to describe an indefinitely small angle $OAQ$; then the triangle $AMm$ will describe a slender pyramid, having its vertex at $A$, and of which the base is equal to a rectangle contained by $Mm$ and $RT$; for the point $M$ moving parallel to the point $R$, it will describe a line equal to that described by $R$, namely to $RT$.

Let $AM\; =\; f$; the angle $MAR\; =\; \backslash theta$; and the angle $OAQ\; =\; d\backslash phi$; then $Mm\; =\; f\; d\backslash theta$; and $TR\; =\; AR\; \backslash times\; d\backslash phi\; =\; f\; \backslash cos.\; \backslash theta\; \backslash cdot\; d\backslash phi$. Wherefore, $B\; =$ base of the pyramid described by $AMm\; =\; d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash cdot\; f^2$; and the direct attraction of the pyramid in the direction $AM\; =\; \backslash frac\{B\}\{f\}\; =\; d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash cdot\; f$. Wherefore, the elementary attraction of the spheroid, in the direction $AC\; =$ direct attraction of the pyramid $\backslash times\; \backslash frac\{AR\}\{AM\}\; \backslash times\; \backslash frac\{AS\}\{AR\}\; =\; d\backslash phi\; \backslash cos.\; \backslash theta\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash cdot\; f$.

Again, let $MR\; =\; x$, $RS\; =\; y$, $CS\; =\; z$, $CP\; =\; k$ and $AC\; =\; k\text{'}$; then (1.)

$$\backslash frac\{x^2\}\{k^2\}\; +\; \backslash frac\{y^2\; +\; z^2\}\{k\text{'}^2\}\; =\; 1.$$

But $x\; =\; f\; \backslash sin.\; \backslash theta$; $y\; =\; AR\; \backslash cdot\; \backslash sin.\; \backslash phi\; =\; f\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash phi$; and $z\; =\; k\text{'}\; -\; f\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash phi$; wherefore, by substitution, we get

$$\backslash frac\{f^2\; \backslash sin.\; \backslash theta^2\}\{k^2\}\; +\; \backslash frac\{f^2\; \backslash cos.\; \backslash theta^2\; \backslash sin.\; \backslash phi^2\}\{k\text{'}^2\}\; +\; \backslash frac\{(k\text{'}\; -\; f\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash phi)^2\}\{k\text{'}^2\}\; =\; 1.$$

From this equation we get

$$f\; =\; 2k\text{'}\; \backslash cdot\; \backslash frac\{\backslash cos.\; \backslash theta\; \backslash cos.\; \backslash phi\}\{\backslash cos.\; \backslash theta^2\; +\; \backslash frac\{k\text{'}^2\}\{k^2\}\; \backslash sin.\; \backslash theta^2\}.$$

Let this value of $f$ be substituted in the expression of the elementary attraction of the spheroid before found, and it will become

$$2k\text{'}\; \backslash cdot\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; \backslash cdot\; \backslash frac\{d\backslash theta\; \backslash cos^2\; \backslash theta\}\{\backslash cos^2\; \backslash theta\; +\; \backslash frac\{k\text{'}^2\}\{k^2\}\; \backslash sin^2\; \backslash theta\}\; :$$

which expression, when integrated from $\backslash varphi\; =\; 0$ to $\backslash varphi\; =\; \backslash pi$, and from $\backslash theta\; =\; 0$ to $\backslash theta\; =\; \backslash frac\{1\}\{2\}\backslash pi$, will give the attraction of half the spheroid: and the double of it, viz.

$$4k\text{'}\; \backslash cdot\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; \backslash cdot\; \backslash frac\{d\backslash theta\; \backslash cos^2\; \backslash theta\}\{\backslash cos^2\; \backslash theta\; +\; \backslash frac\{k\text{'}^2\}\{k^2\}\; \backslash sin^2\; \backslash theta\},$$

being integrated between the same limits, will give the whole attraction of the spheroid.

In the oblate spheroid, $k$ is less than $k\text{'}$: Let $k\text{'}^2\; =\; k^2\; +\; e^2$, and $z\; =\; \backslash sin\; \backslash varphi$: and, by substitution, the element of the attractive force will become

$$4k\text{'}\; \backslash cdot\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; \backslash cdot\; \backslash frac\{dz(1-z^2)\}\{1+e^2z^2\}\; =$$ $$4k\text{'}\; \backslash cdot\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; \backslash cdot\; \backslash frac\{1+e^2\}\{e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash frac\{e\; dz\}\{1+e^2z^2\}\; -\; \backslash frac\{e\; dz\}\{1+e^2\}\; \backslash right\backslash \}.$$

And, by integrating from $z\; =\; 0$ to $z\; =\; 1$, we get

$$4k\text{'}\; \backslash cdot\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; \backslash cdot\; \backslash frac\{1+e^2\}\{e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash text\{arc.\; tan.\; \}\; e\; -\; \backslash frac\{e\}\{1+e^2\}\; \backslash right\backslash \},$$

for the force with which the matter between the sections that contain the angle $OAQ$, attracts the particle $A$ to the centre. But $\backslash int\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; =$

$$\backslash int\; \backslash frac\{d\backslash varphi\}\{2\}\; \backslash cdot\; (1\; +\; \backslash cos^2\; \backslash varphi)\; =\; \backslash frac\{\backslash varphi\}\{2\}\; +\; \backslash frac\{1\}\{4\}\; \backslash sin\; 2\backslash varphi;\; \backslash text\{\; the\; value\; of\}$$

which, between the limits $\backslash varphi\; =\; 0$, and $\backslash varphi\; =\; \backslash pi$, is $=\; \backslash frac\{\backslash pi\}\{2\}$:

wherefore, the attraction of the spheroid on a particle at $A$, is equal to

$$2\backslash pi\; \backslash cdot\; k\text{'}\; \backslash cdot\; \backslash frac\{1+e^2\}\{e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash text\{arc.\; tan.\; \}\; e\; -\; \backslash frac\{e\}\{1+e^2\}\; \backslash right\backslash \}.$$

$$\backslash text\{Because\; \}\; \backslash frac\{4\backslash pi\; \backslash cdot\; k\text{'}^2\; k\}\{3\}\; =\; \backslash frac\{4\backslash pi\; \backslash cdot\; k^2\; \backslash cdot\; (1+e^2)\}\{3\}\; =\; M;\; \backslash text\{\; we\}$$

$$\backslash text\{get\; \}\; 2\backslash pi\; \backslash cdot\; (1+e^2)\; =\; \backslash frac\{3M\}\{2k^2\};\; \backslash text\{\; wherefore,\; the\; measure\}$$

of the attractive force of the oblate spheroid on a particle placed anywhere in the circumference of the circular section made by a plane through the centre at right angles to the axis, is equal to

$$k\text{'}\; \backslash cdot\; \backslash frac\{3M\}\{2k^2\; \backslash cdot\; e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash text\{arc.\; tan.\; \}\; e\; -\; \backslash frac\{e\}\{1+e^2\}\; \backslash right\backslash \}.$$

In the oblong spheroid, $k$ is greater than $k\text{'}$; put $k^2\; =\; k\text{'}^2\; +\; e^2$: then the element of the attractive force will become, by substitution,

$$4k\text{'}\; \backslash cdot\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; \backslash cdot\; \backslash frac\{dz(1-z^2)\}\{1-e^2z^2\}\; =\; 4k\text{'}\; \backslash cdot\; d\backslash varphi\; \backslash cos^2\; \backslash varphi\; \backslash cdot\; \backslash frac\{1-e^2\}\{e^2\}\; \backslash cdot$$ $$\backslash left\backslash \{\; \backslash frac\{e\; dz\}\{1-e^2\}\; -\; \backslash frac\{e\; dz\}\{1-e^2z^2\}\; \backslash right\backslash \}\; :$$

whence, by proceeding as before, we get the measure of the attractive force of the oblong spheroid upon a particle placed anywhere in the circumference of the circular section, made by a plane through the centre at right angles to the axis, equal to

$$k\text{'}\; \backslash cdot\; \backslash frac\{3M\}\{2k^2\; \backslash cdot\; e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash frac\{e\}\{1-e^2\}\; -\; \backslash frac\{1\}\{2\}\; \backslash text\{hyp.\; log.\; \}\; \backslash frac\{1+e\}\{1-e\}\; \backslash right\backslash \}.\; \backslash quad\; \backslash text\{Attraction.\}$$

Cor. In an oblate spheroid, differing little from a sphere, the higher powers of $e^2$ may be neglected. The expression of the attractive force at the equator,

$$\backslash text\{viz.\; \}\; 2\backslash pi\; \backslash cdot\; k\text{'}\; \backslash cdot\; \backslash frac\{1+e^2\}\{e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash text\{arc.\; tan.\; \}\; e\; -\; \backslash frac\{e\}\{1+e^2\}\; \backslash right\backslash \}$$

will then become

$$2\backslash pi\; \backslash cdot\; k\text{'}\; \backslash cdot\; (1+e^2)\; \backslash left(\; \backslash frac\{2\}\{3\}\; -\; \backslash frac\{4\}\{5\}\; e^2\; \backslash right)\; =\; \backslash frac\{4\backslash pi\; k\text{'}\}\{3\}\; \backslash cdot\; (1\; -\; \backslash frac\{1\}\{5\}\; e^2).$$

And if $k\text{'}$, the radius of the equator, $=\; k\; +\; \backslash tau$; then

$$\backslash frac\{2\backslash pi\}\{k\}\; =\; e^2\; \backslash text\{\; (23.\; Cor.)\};\; \backslash text\{\; and\; the\; attraction\; at\; the\; equator\}$$

will be equal to

$$\backslash frac\{4\backslash pi\; k\}\{3\}\; \backslash cdot\; \backslash left(\; 1\; +\; \backslash frac\{\backslash tau\}\{k\}\; \backslash right)\; \backslash cdot\; \backslash left(\; 1\; -\; \backslash frac\{2\}\{5\}\; \backslash cdot\; \backslash frac\{\backslash tau\}\{k\}\; \backslash right)\; =\; \backslash frac\{4\backslash pi\; k\}\{3\}\; \backslash cdot\; \backslash left(\; 1\; +\; \backslash frac\{3\}\{5\}\; \backslash cdot\; \backslash frac\{\backslash tau\}\{k\}\; \backslash right).$$

25. An oblate spheroid of revolution being given, it is required to find the measures of the attractive forces that urge a particle placed anywhere in the surface, in a direction perpendicular to the axis, and in a direction parallel to it.

Let $k$ and $k\text{'}$ be the semiaxes of the ellipse by the revolution of which the spheroid is described, $k$ being the axis about which it revolves: and let $b$ be the perpendicular distance of the particle from the axis, and $a$ its distance from the plane, drawn through the centre at right angles to the axis: Then, from which was proved in (20.) and (21.), the attractions sought will be found by multiplying the attractions at the pole, and at the circular section equally distant from both poles, by $\backslash frac\{a\}\{k\}$ and $\backslash frac\{b\}\{k\text{'}\}$.

Thus we get the attraction in the direction of $a$, equal to

$$a\; \backslash times\; \backslash frac\{3M\}\{k\; \backslash cdot\; e^2\}\; \backslash cdot\; \backslash left\backslash \{\; e\; -\; \backslash text\{arc.\; tan.\; \}\; e\; \backslash right\backslash \};$$

and the attraction in the direction of $b$, equal to

$$b\; \backslash times\; \backslash frac\{3M\}\{2k^2\; \backslash cdot\; e^2\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash text\{arc.\; tan.\; \}\; e\; -\; \backslash frac\{e\}\{1+e^2\}\; \backslash right\backslash \}.$$

The same formulae likewise serve for finding the attractions upon a particle placed anywhere within the spheroid. For the attraction upon a particle within the spheroid is equal to the attraction of a similar concentric spheroid, which contains the particle in its surface (9.); and it is evident, that the coefficients, which multiply $a$ and $b$ in the above expressions, depend only upon the proportion of $k$ and $k\text{'}$; and they are therefore the same for all similar spheroids.

If we denote by $A$ and $B$ the coefficients of $a$ and $b$ in the expressions of the attractive force found above, the whole attraction of the spheroid, which is compounded of the forces $a\; \backslash cdot\; A$ and $b\; \backslash cdot\; B$, will

be $=\; \backslash sqrt\{a^2\; A^2\; +\; b^2\; B^2\}$. And if $\backslash Psi$ denote the angle which the direction of this force makes with $a$, or

with the axis of the spheroid; then $\backslash tan.\; \backslash Psi\; =\; \backslash frac\{b\; \backslash cdot\; B\}\{a\; \backslash cdot\; A\}$.

Attraction. Cor. In the very same manner we may determine the attractions of an oblong spheroid of revolution, upon a point in the surface, or within the solid.

26. If $k,\; k\text{'},\; k\text{'}\text{'}$, the semiaxes of a homogeneous ellipsoid, be related to those of another ellipsoid of the same matter, $h,\; h\text{'},\; h\text{'}\text{'}$, so that $k^2\; -\; k\text{'}^2\; =\; h^2\; -\; h\text{'}^2$ and $k\text{'}^2\; -\; k\text{'}\text{'}^2\; =\; h\text{'}^2\; -\; h\text{'}\text{'}^2$, the attractions perpendicular to the planes of the principal sections, which the first ellipsoid (Plate XXI. fig. 17.) exerts upon a point determined by the coordinates $h\; \backslash sin\; m,\; h\text{'}\; \backslash cos\; m\; \backslash sin\; n,\; h\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; n$, respectively parallel to $k,\; k\text{'},\; k\text{'}\text{'}$, will be to the attractions which the second ellipsoid exerts upon a point determined by the coordinates $k\; \backslash sin\; m,\; k\text{'}\; \backslash cos\; m\; \backslash sin\; n,\; k\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; n$, respectively parallel to $h,\; h\text{'},\; h\text{'}\text{'}$, in the direct proportion of the areas of the principal sections to which the attractions are perpendicular.

This proposition is an extension to all elliptical spheroids of what was proved of the sphere in (13.). It is here enunciated of the ellipsoid, because the demonstration is not more difficult for that solid than for spheroids of revolution.

Let $ABDM$ be an ellipsoid, the semiaxes of which are $BC\; =\; k,\; EC\; =\; k\text{'},\; AC\; =\; k\text{'}\text{'}$; and $acdm$ another ellipsoid, of which the semiaxes are, $bc\; =\; h,\; ec\; =\; h\text{'},\; ac\; =\; h\text{'}\text{'}$; those quantities being so related, that $k^2\; -\; k\text{'}^2\; =\; h^2\; -\; h\text{'}^2$, and $k\text{'}^2\; -\; k\text{'}\text{'}^2\; =\; h\text{'}^2\; -\; h\text{'}\text{'}^2$. Also, let $G$ be a point about the ellipsoid $ABDM$, so determined that $GH$, parallel to $BC$, $=\; h\; \backslash sin\; m$; $HK$, parallel to $CE$, $=\; h\text{'}\; \backslash cos\; m\; \backslash sin\; n$; and $CK\; =\; h\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; n$; and let $g$ be a point about the ellipsoid $acdm$, so determined that $gh$, parallel to $bc$, $=\; k\; \backslash sin\; m$; $hk$, parallel to $ce$, $=\; k\text{'}\; \backslash cos\; m\; \backslash sin\; n$; and $ck\; =\; k\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; n$. Then the force with which the ellipsoid $ABDM$ attracts a particle placed at $G$ in the direction $GH$, will be to the force with which the ellipsoid $acdm$ attracts a particle placed at $g$ in the direction $gh$, as the area of the section $AEDM$ to the area of the section $acdm$, or as $k\text{'}\; k\text{'}\text{'}$ to $h\text{'}\; h\text{'}\text{'}$.

Let $RP\; =\; k\; \backslash sin\; \backslash theta$; $PO\; =\; k\text{'}\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash phi$; and $CO\; =\; k\text{'}\text{'}\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash phi$; which suppositions are allowable, because they satisfy the equation of the ellipsoid (1.), whatever be the angles $\backslash theta$ and $\backslash phi$. Draw $CPM$ through the centre, and $CN$ indefinitely near it; then $CP\; =\; \backslash cos\; \backslash theta\; \backslash sqrt\{k\text{'}^2\; \backslash sin^2\; \backslash phi\; +\; k\text{'}\text{'}^2\; \backslash cos^2\; \backslash phi\}$; and when $\backslash cos\; \backslash theta\; =\; 1$, $CM\; =\; \backslash sqrt\{k\text{'}^2\; \backslash sin^2\; \backslash phi\; +\; k\text{'}\text{'}^2\; \backslash cos^2\; \backslash phi\}$:

wherefore $\backslash frac\{CP\}\{CM\}\; =\; \backslash cos\; \backslash theta$. Let the angle $DCM\; =\; \backslash Psi$;

then $\backslash tan\; \backslash Psi\; =\; \backslash frac\{PO\}\{CO\}\; =\; \backslash frac\{k\text{'}\}\{k\text{'}\text{'}\}\; \backslash tan\; \backslash phi$; and, by taking the

fluxions, $\backslash frac\{d\backslash Psi\}\{\backslash cos^2\; \backslash Psi\}\; =\; \backslash frac\{k\text{'}\}\{k\text{'}\text{'}\}\; \backslash cdot\; \backslash frac\{d\backslash phi\}\{\backslash cos^2\; \backslash phi\}$; but $\backslash frac\{1\}\{\backslash cos^2\; \backslash Psi\}\; =\; 1\; +$

$\backslash tan^2\; \backslash Psi\; =\; \backslash frac\{k\text{'}^2\; \backslash sin^2\; \backslash phi\; +\; k\text{'}\text{'}^2\; \backslash cos^2\; \backslash phi\}\{k\text{'}\text{'}^2\; \backslash cos^2\; \backslash phi\}\; =\; \backslash frac\{CM^2\}\{k\text{'}\text{'}^2\; \backslash cos^2\; \backslash phi\}$;

wherefore $d\backslash Psi\; \backslash cdot\; CM^2\; =$ twice the sector $MCN\; =\; k\text{'}\; k\text{'}\text{'}\; d\backslash phi$.

And, in like manner, in the other ellipsoid, if $rp\; =\; h\; \backslash sin\; \backslash theta$; $po\; =\; h\text{'}\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash phi$; and $co\; =\; h\text{'}\text{'}\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash phi$:

then $\backslash frac\{cp\}\{co\}\; =\; \backslash cos\; \backslash theta$, and twice the sector $mcn\; =\; h\text{'}\; h\text{'}\text{'}\; d\backslash phi$.

It is plain, from what has been shown, that, when Attraction, $\backslash theta$ varies, and $\backslash phi$ remains constant in the expressions of the coordinates, the points $P$ and $p$ will move along

$CM$ and $cm$, so that, in every position, $\backslash frac\{PC\}\{MC\}\; =\; \backslash frac\{pc\}\{mc\}$.

Let $Q$ and $q$ be indefinitely near $P$ and $p$; and through $P$ and $Q$ draw lines parallel to $MN$; and through $p$ and $q$ draw lines parallel to $mn$. Let $S$ denote the quadrilateral contained between the parallels drawn through $P$ and $Q$; and $S\text{'}$ that contained between

the lines drawn through $p$ and $q$: Then $\backslash frac\{S\}\{MCN\}\; =$

$\backslash frac\{QC^2\; -\; PC^2\}\{MC^2\}$; and $\backslash frac\{S\text{'}\}\{MCN\}\; =\; \backslash frac\{qc^2\; -\; pc^2\}\{mc^2\}$; wherefore,

since $\backslash frac\{PC\}\{MC\}\; =\; \backslash frac\{pc\}\{mc\}$, and $\backslash frac\{QC\}\{MC\}\; =\; \backslash frac\{qc\}\{mc\}$, it is manifest that

$\backslash frac\{S\}\{MCN\}\; =\; \backslash frac\{k\text{'}\; k\text{'}\text{'}\}\{h\text{'}\; h\text{'}\text{'}\}$.

Upon the quadrilaterals $S$ and $S\text{'}$ let upright prisms $RS$ and $rs$ be erected, and be prolonged to meet the surfaces of the spheroids; join $GR,\; GS,\; gr,\; gs$. Then,

$GR^2\; =\; (h\; \backslash sin\; m\; -\; k\; \backslash sin\; \backslash theta)^2\; +\; (h\text{'}\; \backslash cos\; m\; \backslash sin\; n\; -\; k\text{'}\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash phi)^2$;

$+\; (h\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; n\; -\; k\text{'}\text{'}\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash phi)^2$;

$gr^2\; =\; (k\; \backslash sin\; m\; -\; h\; \backslash sin\; \backslash theta)^2\; +\; (k\text{'}\; \backslash cos\; m\; \backslash sin\; n\; -\; h\text{'}\; \backslash cos\; \backslash theta\; \backslash sin\; \backslash phi)^2$;

$+\; (k\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; n\; -\; h\text{'}\text{'}\; \backslash cos\; \backslash theta\; \backslash cos\; \backslash phi)^2$;

And, by expanding these expressions, we get

$GR^2\; =\; \backslash left\backslash \{\; (h^2\; +\; (k\text{'}^2\; -\; k^2)\; \backslash cos^2\; m\; \backslash sin^2\; n\; +\; (h\text{'}\text{'}^2\; -\; k\text{'}\text{'}^2)\; \backslash cos^2\; m\; \backslash cos^2\; n)\; \backslash right.$

$+\; (k^2\; +\; (k\text{'}^2\; -\; k^2)\; \backslash cos^2\; \backslash theta\; \backslash sin^2\; \backslash phi\; +\; (k\text{'}\text{'}^2\; -\; k\text{'}\text{'}^2)\; \backslash cos^2\; \backslash theta\; \backslash cos^2\; \backslash phi)$

$-\; 2(hk\; \backslash sin\; m\; \backslash sin\; \backslash theta\; +\; h\text{'}k\text{'}\; \backslash cos\; m\; \backslash cos\; \backslash theta\; \backslash sin\; n\; \backslash sin\; \backslash phi\; +\; h\text{'}\text{'}k\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; \backslash theta\; \backslash cos\; n\; \backslash cos\; \backslash phi)\; \backslash left.\; \backslash right\backslash \}$;

$gr^2\; =\; \backslash left\backslash \{\; (k^2\; +\; (k\text{'}^2\; -\; k^2)\; \backslash cos^2\; m\; \backslash sin^2\; n\; +\; (k\text{'}\text{'}^2\; -\; k\text{'}\text{'}^2)\; \backslash cos^2\; m\; \backslash cos^2\; n)\; \backslash right.$

$+\; (h^2\; +\; (h\text{'}^2\; -\; h^2)\; \backslash cos^2\; \backslash theta\; \backslash sin^2\; \backslash phi\; +\; (h\text{'}\text{'}^2\; -\; h\text{'}\text{'}^2)\; \backslash cos^2\; \backslash theta\; \backslash cos^2\; \backslash phi)$

$-\; 2(hk\; \backslash sin\; m\; \backslash sin\; \backslash theta\; +\; h\text{'}k\text{'}\; \backslash cos\; m\; \backslash cos\; \backslash theta\; \backslash sin\; n\; \backslash sin\; \backslash phi\; +\; h\text{'}\text{'}k\text{'}\text{'}\; \backslash cos\; m\; \backslash cos\; \backslash theta\; \backslash cos\; n\; \backslash cos\; \backslash phi)\; \backslash left.\; \backslash right\backslash \}$.

These expressions are equal, because $k\text{'}^2\; -\; k^2\; =\; h\text{'}^2\; -\; h^2$, and $k\text{'}\text{'}^2\; -\; k^2\; =\; h\text{'}\text{'}^2\; -\; h^2$: wherefore $RG\; =\; rg\; =\; f$. And, in like manner, it is shown, that $GS\; =\; gs\; =\; f\text{'}$.

Now, the attraction of the prism $RS$ urging a particle at $G$ in the direction $GH$, is equal to

$S\; \backslash times\; \backslash left\backslash \{\; \backslash frac\{1\}\{f\}\; -\; \backslash frac\{1\}\{f\text{'}\}\; \backslash right\backslash \}$ (10); and the attraction of the prism $rs$ urging a particle at $g$ in the direction $gh$,

is $S\text{'}\; \backslash times\; \backslash left\backslash \{\; \backslash frac\{1\}\{f\}\; -\; \backslash frac\{1\}\{f\text{'}\}\; \backslash right\backslash \}$: wherefore these attractions

Attraction. are to one another as $S$ to $S\text{'}$, or as $k\text{'}k\text{'}\text{'}$ to $h\text{'}h\text{'}\text{'}$. The same thing may be proved of all the elementary prisms that make up the two portions of the spheroids contained between the planes $BCM$, $BCN$, and $bcn$, $bcn\text{'}$: wherefore, those portions attract particles at $G$ and $g$, with forces proportional to $k\text{'}k\text{'}\text{'}$ and $h\text{'}h\text{'}\text{'}$. But the two spheroids may be divided into an equal number of such portions: wherefore the spheroids attract particles placed at $G$ and $g$, in the directions $GH$ and $gh$, with forces proportional to $k\text{'}k\text{'}\text{'}$ and $h\text{'}h\text{'}\text{'}$, or to the sections $AMDE$ and $amde$.

Cor. 1. This proposition is true when the law of attraction is expressed by any function of the distance. The demonstration is the same as in the corollary of (13).

Cor. 2. If the two ellipsoids be so placed, that their centres, and the planes of their principal sections, shall coincide, the surface of the one will be entirely within the other. Also the point which one ellipsoid attracts, will be in the surface of the other, as is plain from the expressions of the coordinate. And hence, the attraction of one ellipsoid upon a point without the surface, is made to depend upon the attraction of another ellipsoid upon a point within the surface.

Cor. 3. When the ellipsoids become spheroids of revolution, the two principal sections through the axis of revolution become equal, and will be represented by any two sections whatever passing through the axis at right angles to one another. But, in this case, the attractions of the spheroids on the points may be reduced to two, one acting perpendicular to the axis, and one parallel to it: And it is plain, that these attractions will be to one another as the areas of the sections, perpendicular to their directions.

27. To find the attraction of an oblate spheroid upon a particle placed without the surface.

Let $k\text{'}$ be the radius of the equator, and $k$ the axis of revolution: and let $a$ be the perpendicular distance of the point without the spheroid from the plane of the equator, and $b$ its distance from the axis. In the first place, it is necessary to determine the semiaxis of another oblate spheroid that shall contain the given point in its surface, and such, that it shall have the same centre, and its equator in the same plane, as the given spheroid; and likewise, the difference of the squares of its semiaxes equal to the difference of the squares of the semiaxes of the given spheroid. Let $h\text{'}$ denote the radius of the equator, and $h$ the semiaxis of the required spheroid: then, because the attracted point is to be in the surface of

the solid, we have $\backslash frac\{a^2\}\{h^2\}\; +\; \backslash frac\{b^2\}\{h\text{'}^2\}\; =\; 1$: and, because

$$h\text{'}^2\; -\; k^2\; =\; k\text{'}^2\; -\; k^2\; =\; i^2;\; \backslash text\{\; we\; get\}$$

$$\backslash frac\{a^2\}\{k^2\}\; +\; \backslash frac\{b^2\}\{k\text{'}^2\}\; =\; 1:$$

Whence,

$$2k^2\; =\; a^2\; +\; b^2\; -\; i^2\; +\; \backslash sqrt\{(a^2\; +\; b^2\; -\; i^2)^2\; +\; 4a^2i^2\};$$

and when $h$ is determined, then $h\text{'}\; =\; \backslash sqrt\{k^2\; +\; i^2\}$.

In consequence of the equation $\backslash frac\{a^2\}\{h^2\}\; +\; \backslash frac\{b^2\}\{h\text{'}^2\}\; =\; 1$, Attraction.

we may suppose, $a\; =\; h\; \backslash sin.\; m$, and $b\; =\; h\text{'}\; \backslash cos.\; m$;

let $a\text{'}\; =\; k\; \backslash sin.\; m$, and $b\text{'}\; =\; k\text{'}\; \backslash cos.\; m$; or $a\text{'}\; =\; \backslash frac\{k\}\{h\}\; a$,

and $b\text{'}\; =\; \backslash frac\{k\text{'}\}\{h\text{'}\}\; \backslash times\; b$: then the point determined by the

coordinates $a\text{'}$ and $b\text{'}$ will be in the surface of the given spheroid, and, consequently, it will be within the surface of the other spheroid. Let $M\text{'}$ denote the mass of the spheroid of which the axis is $h$; and let

$$e\text{'}^2\; =\; \backslash frac\{h\text{'}^2\; -\; k^2\}\{h^2\}\; =\; \backslash frac\{k\text{'}^2\; -\; k^2\}\{h^2\};\; \backslash text\{\; then\; (25.)\; the\; attractions\}$$

of this spheroid upon the point within its surface, determined by the coordinates $a$ and $b$, are these, viz.

That perpendicular to the equator, equal to

$$a\text{'}\; \backslash times\; \backslash frac\{3M\text{'}\}\{h^3\; e\text{'}^3\}\; \backslash cdot\; \backslash left\backslash \{\; e\text{'}\; -\; \backslash text\{arc.\; tan.\; \}\; e\text{'}\; \backslash right\backslash \};$$

and that perpendicular to the axis, equal to

$$b\text{'}\; \backslash times\; \backslash frac\{3M\text{'}\}\{2h^3\; e\text{'}^3\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash text\{arc.\; tan.\; \}\; e\text{'}\; -\; \backslash frac\{e\text{'}\}\{1\; +\; e\text{'}^2\}\; \backslash right\backslash \}.$$

But (26. Cor. 3.) the attractions of the given spheroid, whose semiaxis are $k$ and $k\text{'}$ upon the point without its surface determined by the coordinates $a$ and $b$, will be found by multiplying the preceding expressions respectively by $\backslash frac\{k\text{'}^2\}\{k^2\}$ and $\backslash frac\{kk\text{'}\}\{hh\text{'}\}$. Let $M$ be the

mass of the given spheroid; then $\backslash frac\{M\}\{M\text{'}\}\; =\; \backslash frac\{k\text{'}^2\; k\}\{h^2\; h\text{'}\}$; conse-

quently $\backslash frac\{k\text{'}^2\}\{k^2\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash cdot\; \backslash frac\{h\}\{k\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash cdot\; \backslash frac\{a\}\{a\text{'}\}$; and $\backslash frac\{kk\text{'}\}\{hh\text{'}\}\; =$

$\backslash frac\{M\}\{M\text{'}\}\; \backslash cdot\; \backslash frac\{h\text{'}\}\{k\text{'}\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash cdot\; \backslash frac\{b\}\{b\text{'}\}$: Wherefore, the attractions of the given oblate spheroid upon a point, without the surface determined by the coordinates $a$ and $b$, are as follows, viz.

The attraction perpendicular to the equator, equal to

$$a\; \backslash times\; \backslash frac\{3M\}\{h^3\; e\text{'}^3\}\; \backslash cdot\; \backslash left\backslash \{\; e\text{'}\; -\; \backslash text\{arc.\; tan.\; \}\; e\text{'}\; \backslash right\backslash \};$$

and that perpendicular to the axis, equal to

$$b\; \backslash times\; \backslash frac\{3M\}\{2h^3\; e\text{'}^3\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash text\{arc.\; tan.\; \}\; e\text{'}\; -\; \backslash frac\{e\text{'}\}\{1\; +\; e\text{'}^2\}\; \backslash right\backslash \}.$$

Cor. In the very same manner we may determine the attractions of an oblong spheroid of revolution upon a point without the surface.

ATTRACTIONS OF ELLIPSOIDS.

28. Let $AMBN$ be one of the principal sections of Attractions an ellipsoid, $C$ the centre, $AB$ and $MN$ the axes, $D$ a point in the periphery of the section, and $DO$ perpendicular to $MN$ (Plate XXXI. fig. 18.); the attraction of the ellipsoid upon a particle placed at the pole $A$, is to the force with which a particle placed at $D$ is attracted in the direction $DO$, as $AC$ to $DO$.

Attraction. Draw DFG perpendicular to AB, and through F describe an ellipsoid similar to the given ellipsoid, and similarly situated, and having the same centre. Conceive an indefinitely great number of planes making indefinitely small angles with one another, to be drawn through DG, so as to divide the two ellipsoids into an indefinitely great number of thin solids or slices: Then, the sections of the ellipsoids made by every one of the planes will be similar and concentric ellipses, each of them having an axis perpendicular to DG (5.). Wherefore the attractions of the elements of the ellipsoid FHKL upon a particle at F, are respectively equal to the attractions of the elements of the ellipsoid AMBN, upon a particle at D in the direction DO (19.). Wherefore, the whole attraction of the ellipsoid FHKL upon a particle at F, is equal to the attraction of the ellipsoid AMBN upon a particle at D, in the direction DO. But the attractions of the ellipsoids AMBN, and FHKL upon particles at A and F, are to one another as AC to CF (8.). Wherefore, the attraction of the ellipsoid AMBN upon a particle at the pole A, is to the force with which it attracts a particle at D in the direction DO, as AC to DO.

29. The attractions of ellipsoids upon particles placed in the surface, urging them in directions perpendicular to any of the principal sections, are proportional to the distances of the particles from that section.

Let AMBN be one of the principal sections of an ellipsoid, C the centre, AB and MN the axes of the section, and P a point in the surface of the solid; the attraction of the ellipsoid upon a particle at the pole A (Plate XXXI. fig. 19.), is to the force with which a particle at P is attracted in a direction parallel to AB, as the semiaxis AC is to the distance of P from the principal section perpendicular to AC.

Draw PDQ perpendicular to the section AMBN, and let it meet the surface again in Q; through D describe an ellipsoid similar to AMBN, similarly situated, and having the same centre; and through P draw a section SPRQ perpendicular to AB. As before, divide the solids into an indefinitely great number of thin slices by planes drawn through PQ: the sections made by every one of those planes will be similar, and concentric ellipses having an axis of each perpendicular to PQ (5.). Wherefore, the attractions of the elements of the ellipsoid AMBN, upon a particle at P, in a direction perpendicular to the plane PRQS, are respectively equal to the attractions of the elements of the ellipsoid FHKL, upon a particle at D, in a direction perpendicular to the same plane (19.). Wherefore, the attraction of the ellipsoid AMBN, upon a particle at P, in a direction parallel to the axis AB, is equal to the attraction of the ellipsoid FHKL, upon a particle at D in the same direction. But the ellipsoids AMBN and FHKL being similar, their attractions upon particles at A and F, are to one another as AC to CF (8.); and the attraction of the ellipsoid FHKL, upon a particle at the pole F, is to its attraction upon a particle at D, in a direction parallel to AC, as FC to CN (28.). Wherefore (ex aequali), the attraction of the ellipsoid AMBN, upon a particle, at the pole A, is to the force with which it attracts a particle at P, in the direction AC, as AC to CN.

This proposition will enable us to find the attraction.

VOL. I. PART II.

tions of an ellipsoid on all points on the surface, or Attraction. within the solid, when the attractions at the poles are determined.

30. To investigate the differential expressions of the attractions at the poles of an ellipsoid.

Let APD be an ellipsoid; C the centre; AC, CE, and PC, the semiaxes; and PMB a section made by a plane through PC and any point, M in the surface: draw PM (fig. 20.) $Pm$ indefinitely near PM, and $Mm$ perpendicular to $Pm$: also MR perpendicular to the plane ADB, MS perpendicular to PC, and RH perpendicular to AD. Conceive the plane PCB to revolve about PC, so as to describe an indefinitely small angle BCO; and let $PM\; =\; f$; the angle KPM, which PM makes with a perpendicular to the axis, $=\; \backslash theta$; and the angle DCB $=\; \backslash varphi$: then by proceeding as in (23), it will be found that the attraction of the small pyramid described by the triangle $MPm$, urging a particle at P to the centre of the ellipsoid, is $=\; d\backslash varphi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash theta\; \backslash cdot\; f$.

Again, let $MR\; =\; x$, $HR\; =\; y$, $CH\; =\; z$; also let $PC\; =\; k$, $AC\; =\; k\text{'}$, $CE\; =\; k\text{'}\text{'}$; then $x\; =\; k\; \backslash sin.\; \backslash theta$; $y\; =\; f\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi$; $z\; =\; f\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi$; and if we substitute these values in the equation of the ellipsoid (1.), we shall get

$$\backslash frac\{(k\; -\; f\; \backslash sin.\; \backslash theta)^2\}\{k^2\}\; +\; \backslash frac\{f^2\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\}\{k\text{'}^2\}\; +\; \backslash frac\{f^2\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}\{k\text{'}\text{'}^2\}$$

$=\; 1$: whence

$$f\; =\; \backslash frac\{k\}\{k^2\}\; \backslash cdot\; \backslash frac\{2\; \backslash sin.\; \backslash theta\}\{\backslash frac\{\backslash sin.\; \backslash theta\}\{k^2\}\; +\; \backslash frac\{\backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\}\{k\text{'}^2\}\; +\; \backslash frac\{\backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}\{k\text{'}\text{'}^2\}\}$$

This is the value of $f$ at the pole of $k$; and, by a like procedure, its values at the poles of $k\text{'}$ and $k\text{'}\text{'}$ may be found, viz.

$$f\; =\; \backslash frac\{k\text{'}\}\{k\text{'}^2\}\; \backslash cdot\; \backslash frac\{2\; \backslash sin.\; \backslash theta\}\{\backslash frac\{\backslash sin.\; \backslash theta\}\{k^2\}\; +\; \backslash frac\{\backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\}\{k^2\}\; +\; \backslash frac\{\backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}\{k\text{'}\text{'}^2\}\}$$

$$f\; =\; \backslash frac\{k\text{'}\text{'}\}\{k\text{'}\text{'}^2\}\; \backslash cdot\; \backslash frac\{2\; \backslash sin.\; \backslash theta\}\{\backslash frac\{\backslash sin.\; \backslash theta\}\{k^2\}\; +\; \backslash frac\{\backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\}\{k^2\}\; +\; \backslash frac\{\backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}\{k^2\}\}$$

Suppose that $k$ is the least of the semiaxes; and let $k^2\; =\; \backslash frac\{k^2\}\{m\}$, and $k\text{'}^2\; =\; \backslash frac\{k^2\}\{n\}$: then the values of $f$ at the poles of $k$, $k\text{'}$, $k\text{'}\text{'}$, will be, respectively,

$$f\; =\; \backslash frac\{2k\; \backslash sin.\; \backslash theta\}\{\backslash sin.\; \backslash theta\; +\; m\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\; +\; n\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}$$

$$f\; =\; \backslash frac\{2k\text{'}\; \backslash sin.\; \backslash theta\; \backslash times\; m\}\{m\; \backslash sin.\; \backslash theta\; +\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\; +\; n\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}$$

$$f\; =\; \backslash frac\{2k\text{'}\text{'}\; \backslash sin.\; \backslash theta\; \backslash times\; n\}\{n\; \backslash sin.\; \backslash theta\; +\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\; +\; m\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}$$

Now, let $A$, $A\text{'}$, $A\text{'}\text{'}$, denote the attractions of the spheroid upon particles placed at the poles of $k$, $k\text{'}$, $k\text{'}\text{'}$; then, by substituting the values of $f$ just found in the foregoing differential expression, we get

$$A\; =\; k\; \backslash times\; \backslash iint\; \backslash frac\{2\; d\backslash varphi\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash theta\}\{\backslash sin.\; \backslash theta\; +\; m\; \backslash cos.\; \backslash theta\; \backslash sin.\; \backslash varphi\; +\; n\; \backslash cos.\; \backslash theta\; \backslash cos.\; \backslash varphi\}$$

$A$ M

$$\backslash text\{Attraction.\}\; \backslash quad\; A\text{'}\; =\; k\text{'}\; \backslash times\; \backslash iint\; \backslash frac\{m\; \backslash times\; 2\; d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; 2\backslash theta\}\{m\; \backslash sin.\; 2\backslash theta\; +\; \backslash cos.\; 2\backslash theta\; \backslash sin.\; 2\backslash phi\; +\; n\; \backslash cos.\; 2\backslash theta\; \backslash cos.\; 2\backslash phi\}$$

$$A\text{'}\text{'}\; =\; k\text{'}\text{'}\; \backslash times\; \backslash iint\; \backslash frac\{n\; \backslash times\; 2\; d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; 2\backslash theta\}\{n\; \backslash sin.\; 2\backslash theta\; +\; \backslash cos.\; 2\backslash theta\; \backslash sin.\; 2\backslash phi\; +\; m\; \backslash cos.\; 2\backslash theta\; \backslash cos.\; 2\backslash phi\}$$

the limits of the integrals being from $\backslash theta\; =\; 0$ and $\backslash phi\; =\; 0$ to $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{2\}$ and $\backslash phi\; =\; 2\backslash pi$.

31. To reduce the expressions of the polar attractions to the most simple integrals.

Let us consider the general expression

$$\backslash iint\; \backslash frac\{d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; 2\backslash theta\}\{\backslash alpha\; \backslash sin.\; 2\backslash theta\; +\; \backslash beta\; \backslash cos.\; 2\backslash theta\; \backslash sin.\; 2\backslash phi\; +\; \backslash gamma\; \backslash cos.\; 2\backslash theta\; \backslash cos.\; 2\backslash phi\}$$

which includes all the formulas found in (30.) Let $p\; =\; \backslash alpha\; \backslash sin.\; 2\backslash theta\; +\; \backslash beta\; \backslash cos.\; 2\backslash theta$; and $q\; =\; \backslash alpha\; \backslash sin.\; 2\backslash theta\; +\; \backslash gamma\; \backslash cos.\; 2\backslash theta$; then the above expression will become

$$\backslash iint\; \backslash frac\{d\backslash phi\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; 2\backslash theta\}\{p\; \backslash sin.\; 2\backslash phi\; +\; q\; \backslash cos.\; 2\backslash phi\}.$$

Suppose $\backslash sqrt\{\backslash frac\{q\}\{p\}\}\; \backslash cdot\; \backslash frac\{\backslash sin.\; \backslash phi\}\{\backslash cos.\; \backslash phi\}\; =\; \backslash frac\{\backslash sin.\; u\}\{\backslash cos.\; u\}$; then the preceding expression will become, by substitution,

$$\backslash iint\; \backslash frac\{du\; \backslash cdot\; d\backslash theta\; \backslash cos.\; \backslash theta\; \backslash sin.\; 2\backslash theta\}\{\backslash sqrt\{p\; \backslash cdot\; q\}\};$$

the limits of $u$ being from 0 to $2\backslash pi$; wherefore, by integrating with regard to $u$, and restoring the values of $p$ and $q$, the integral becomes

$$2\backslash pi\; \backslash cdot\; \backslash int\; \backslash frac\{d\backslash theta\; \backslash cdot\; \backslash cos.\; \backslash theta\; \backslash sin.\; 2\backslash theta\}\{\backslash sqrt\{(\backslash alpha\; \backslash sin.\; 2\backslash theta\; +\; \backslash beta\; \backslash cos.\; 2\backslash theta)\; \backslash cdot\; (\backslash alpha\; \backslash sin.\; 2\backslash theta\; +\; \backslash gamma\; \backslash cos.\; 2\backslash theta)\}\}$$

and, by putting $x\; =\; \backslash sin.\; \backslash theta$, the integral, which is to be taken from $\backslash theta\; =\; 0$ to $\backslash theta\; =\; \backslash frac\{\backslash pi\}\{2\}$, or from $x\; =\; 0$ to $x\; =\; 1$, will become

$$2\backslash pi\; \backslash cdot\; \backslash int\; \backslash frac\{x^2\; dx\}\{\backslash sqrt\{\backslash \{\backslash beta\; +\; (\backslash alpha\; -\; \backslash beta)x^2\backslash \}\; \backslash cdot\; \backslash \{\backslash gamma\; +\; (\backslash alpha\; -\; \backslash gamma)x^2\backslash \}\}\}$$

If now we take $\backslash alpha,\; \backslash beta,\; \backslash gamma$, so as to make the assumed expression coincide with the quantities $A,\; A\text{'},\; A\text{'}\text{'}$, respectively, we shall get

$$A\; =\; 4\backslash pi\; k\; \backslash cdot\; \backslash int\; \backslash frac\{x^2\; dx\}\{\backslash sqrt\{\backslash \{m\; +\; (1-m)x^2\backslash \}\; \backslash cdot\; \backslash \{n\; +\; (1-n)x^2\backslash \}\}\}$$

$$A\text{'}\; =\; 4\backslash pi\; k\text{'}\; \backslash cdot\; \backslash int\; \backslash frac\{m\; \backslash cdot\; x^2\; dx\}\{\backslash sqrt\{\backslash \{1\; +\; (m-1)x^2\backslash \}\; \backslash cdot\; \backslash \{n\; +\; (m-n)x^2\backslash \}\}\}$$

$$A\text{'}\text{'}\; =\; 4\backslash pi\; k\text{'}\text{'}\; \backslash cdot\; \backslash int\; \backslash frac\{n\; \backslash cdot\; x^2\; dx\}\{\backslash sqrt\{\backslash \{1\; +\; (n-1)x^2\backslash \}\; \backslash cdot\; \backslash \{m\; +\; (n-m)x^2\backslash \}\}\}$$

These expressions have the inconvenience of containing different factors in the denominators; but they may be reduced to others having the same factors, by putting $x\; =\; \backslash frac\{\backslash tau\}\{\backslash sqrt\{m\; +\; (1-m)\backslash tau^2\}\}$ in the second and $x\; =$

$\backslash frac\{\backslash tau\}\{\backslash sqrt\{n\; +\; (1-n)\backslash tau^2\}\}$ in the third; we thus get

$$A\; =\; 4\backslash pi\; k\; \backslash cdot\; \backslash int\; \backslash frac\{x^2\; dx\}\{\backslash sqrt\{\backslash \{m\; +\; (1-m)x^2\backslash \}\; \backslash cdot\; \backslash \{n\; +\; (1-n)x^2\backslash \}\}\}\; \backslash quad\; \backslash text\{Attraction.\}$$

$$A\text{'}\; =\; 4\backslash pi\; k\text{'}\; \backslash cdot\; \backslash int\; \backslash frac\{m\; \backslash cdot\; \backslash tau^2\; d\backslash tau\}\{\backslash sqrt\{\backslash \{m\; +\; (1-m)\backslash tau^2\backslash \}\; \backslash cdot\; \backslash \{n\; +\; (1-n)\backslash tau^2\backslash \}\}\}$$

$$A\text{'}\text{'}\; =\; 4\backslash pi\; k\text{'}\text{'}\; \backslash cdot\; \backslash int\; \backslash frac\{n\; \backslash cdot\; \backslash tau^2\; d\backslash tau\}\{\backslash sqrt\{\backslash \{n\; +\; (1-n)\backslash tau^2\backslash \}\; \backslash cdot\; \backslash \{m\; +\; (1-m)\backslash tau^2\backslash \}\}\}$$

Now let

$$\backslash frac\{1-m\}\{m\}\; =\; \backslash frac\{k^2\; -\; k\text{'}^2\}\{k^2\}\; =\; \backslash lambda^2;\; \backslash text\{\; and\; \}\; \backslash frac\{1-n\}\{n\}\; =\; \backslash frac\{k\text{'}^2\; -\; k^2\}\{k^2\}\; =\; \backslash lambda\text{'}^2;$$

Also let the mass of the ellipsoid = $M\; =\; \backslash frac\{4\backslash pi\; k\; k\text{'}\; k\text{'}\text{'}\}\{3\}$; then $\backslash frac\{3M\}\{k^3\}\; =\; \backslash frac\{4\backslash pi\}\{\backslash sqrt\{mn\}\}$; wherefore, by substitution, we get

$$A\; =\; k\; \backslash cdot\; \backslash frac\{3M\}\{k^3\}\; \backslash cdot\; \backslash int\; \backslash frac\{x^2\; dx\}\{\backslash sqrt\{(1\; +\; \backslash lambda^2\; x^2)\; \backslash cdot\; (1\; +\; \backslash lambda\text{'}^2\; x^2)\}\}$$

$$A\text{'}\; =\; k\text{'}\; \backslash cdot\; \backslash frac\{3M\}\{k^3\}\; \backslash cdot\; \backslash int\; \backslash frac\{x^2\; dx\}\{\backslash sqrt\{\backslash \{1\; +\; \backslash lambda^2\; x^2\backslash \}\; \backslash cdot\; \backslash \{1\; +\; \backslash lambda\text{'}^2\; x^2\backslash \}\}\}$$

$$A\text{'}\text{'}\; =\; k\text{'}\text{'}\; \backslash cdot\; \backslash frac\{3M\}\{k^3\}\; \backslash cdot\; \backslash int\; \backslash frac\{x^2\; dx\}\{\backslash sqrt\{\backslash \{1\; +\; \backslash lambda^2\; x^2\backslash \}\; \backslash cdot\; \backslash \{1\; +\; \backslash lambda\text{'}^2\; x^2\backslash \}\}\}$$

the integrations extending from $x\; =\; 0$ to $x\; =\; 1$.

These integrals cannot be expressed in finite terms. When $\backslash lambda$ and $\backslash lambda\text{'}$, or the eccentricities of the ellipsoid are small, the values of the integrals may easily be found to a sufficient degree of exactness, by series. They may likewise be all expressed by means of this fluent, viz.

$$F\; =\; \backslash int\; \backslash frac\{dx\}\{\backslash sqrt\{(1\; +\; \backslash lambda^2\; x^2)\; \backslash cdot\; (1\; +\; \backslash lambda\text{'}^2\; x^2)\}\}\; \backslash quad\; (\backslash text\{from\; \}\; x\; =\; 0\; \backslash text\{\; to\; \}\; x\; =\; 1)\; \backslash text\{\; and\; its\; partial\; fluxions.\; Thus\; we\; have,\; in\; general,\}$$

$$\backslash frac\{x^2\}\{\backslash sqrt\{(1\; +\; \backslash lambda^2\; x^2)\; \backslash cdot\; (1\; +\; \backslash lambda\text{'}^2\; x^2)\}\}\; =\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash cdot\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right)\; +\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash cdot\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right)$$

Wherefore, making $x\; =\; 1$, we get

$$A\; =\; k\; \backslash cdot\; \backslash frac\{3M\}\{k^3\}\; \backslash cdot\; \backslash left\backslash \{\; \backslash frac\{1\}\{\backslash sqrt\{(1\; +\; \backslash lambda^2)\; \backslash cdot\; (1\; +\; \backslash lambda\text{'}^2)\}\}\; +\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right)\; +\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right)\; \backslash right\backslash \}$$

$$A\text{'}\; =\; k\text{'}\; \backslash cdot\; \backslash frac\{3M\}\{k^3\}\; \backslash cdot\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash cdot\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right).$$

$$A\text{'}\text{'}\; =\; k\text{'}\text{'}\; \backslash cdot\; \backslash frac\{3M\}\{k^3\}\; \backslash cdot\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash cdot\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right).$$

32. To find the forces with which a homogeneous ellipsoid attracts a particle placed in the surface, or within the solid, in directions perpendicular to the principal sections.

Let $k,\; k\text{'},\; k\text{'}\text{'}$, denote the semiaxes of an ellipsoid,

Attraction, and $a,\; b,\; c$ (respectively parallel to $k,\; k\text{'},\; k\text{'}\text{'}$), the perpendicular distances of a particle placed in the surface, or within the solid, from the principal sections: then, from what is proved in (29.), the attractions we are seeking will be found by multiplying the polar attractions by $\backslash frac\{a\}\{k\},\; \backslash frac\{b\}\{k\text{'}\},\; \backslash frac\{c\}\{k\text{'}\text{'}\}$. Wherefore the forces that urge the particle in the directions of $a,\; b$, and $c$, are respectively, $a\; \backslash times\; \backslash frac\{3M\}\{k^3\}$.

$$\backslash left\backslash \{\; \backslash frac\{1\}\{\backslash sqrt\{(1+\backslash lambda^2)(1+\backslash lambda\text{'}^2)\}\}\; +\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right)\; +\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right)\; \backslash right\backslash \};$$ $$b\; \backslash times\; \backslash frac\{3M\}\{k\text{'}^3\}\; \backslash cdot\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right);$$ $$c\; \backslash times\; \backslash frac\{3M\}\{k\text{'}\text{'}^3\}\; \backslash cdot\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right);$$

Which formulas serve both for points in the surface, and within the solid, for the reason already explained in (25.).

33. To find the attractions of an ellipsoid upon a particle placed without the surface.

Let $k,\; k\text{'},\; k\text{'}\text{'}$ be the semiaxes of the ellipsoid, and $a,\; b,\; c$ (respectively parallel to $k,\; k\text{'},\; k\text{'}\text{'}$), the coordinates of a particle without the surface. Let $h,\; h\text{'},\; h\text{'}\text{'}$, so related to $k,\; k\text{'},\; k\text{'}\text{'}$, that $k^2\; -\; h^2\; =\; k\text{'}^2\; -\; h\text{'}^2$ and $k\text{'}\text{'}^2\; -\; h\text{'}\text{'}^2\; =\; k\text{'}^2\; -\; h\text{'}^2$, denote the semiaxes of another ellipsoid, which contains the attracted point in its surface, and has its principal sections in the same planes as the given ellipsoid: then, because the attracted point is in the surface, we have (1.)

$$\backslash frac\{a^2\}\{h^2\}\; +\; \backslash frac\{b^2\}\{h\text{'}^2\}\; +\; \backslash frac\{c^2\}\{h\text{'}\text{'}^2\}\; =\; 1:$$

And, because $h^2\; -\; h\text{'}^2\; =\; k^2\; -\; k\text{'}^2\; =\; i^2$, and $h\text{'}\text{'}^2\; -\; h\text{'}^2\; =\; k\text{'}\text{'}^2\; -\; k\text{'}^2\; =\; i^2$, we get

$$\backslash frac\{a^2\}\{h^2\}\; +\; \backslash frac\{b^2\}\{h^2\; +\; i^2\}\; +\; \backslash frac\{c^2\}\{h^2\; +\; i^2\}\; =\; 1.$$

This equation now contains only one unknown quantity; and it is plain, that one value of $h$, and only one, can be found from it. For, when $h\; =\; 0$, the function on the left hand side is infinitely great: And while $h$ increases from 0 ad infinitum, the same function decreases continually from being infinitely great, to be infinitely little. When $h$ is found, then $h\text{'}\; =\; \backslash sqrt\{h^2\; +\; i^2\}$, and $h\text{'}\text{'}\; =\; \backslash sqrt\{h^2\; +\; i^2\}$. Because $a,\; b,\; c$, are the coordinates of a point in the surface of the ellipsoid, we may suppose $a\; =\; h\; \backslash sin.\; m,\; b\; =\; h\text{'}\; \backslash cos.\; m\; \backslash sin.\; n,\; c\; =\; h\text{'}\text{'}\; \backslash cos.\; m\; \backslash cos.\; n$: let $a\text{'}\; =\; k\; \backslash sin.\; m,\; b\text{'}\; =\; k\text{'}\; \backslash cos.\; m\; \backslash sin.\; n,\; c\text{'}\; =\; k\text{'}\text{'}\; \backslash cos.\; m\; \backslash cos.\; n$; or $a\text{'}\; =\; \backslash frac\{k\}\{h\}\; \backslash times\; a$,

$$b\text{'}\; =\; \backslash frac\{k\text{'}\}\{h\text{'}\}\; \backslash times\; b,\; c\text{'}\; =\; \backslash frac\{k\text{'}\text{'}\}\{h\text{'}\text{'}\}\; \backslash times\; c;\; \backslash text\{\; then\; \}\; a\text{'},\; b\text{'},\; c\text{'},\; \backslash text\{\; will\; be\; the\}$$

coordinates of a point in the surface of the given ellipsoid, and consequently, it will be within the other solid. Let $M\text{'}$ denote the mass of the ellipsoid of

which $h,\; h\text{'},\; h\text{'}\text{'}$ are the semiaxes; also let $\backslash lambda^2\; =\; \backslash frac\{k^2\; -\; h^2\}\{k^2\}\; =\; \backslash frac\{k\text{'}^2\; -\; h\text{'}^2\}\{k\text{'}^2\}$; and $\backslash lambda\text{'}^2\; =\; \backslash frac\{h\text{'}^2\; -\; h^2\}\{h\text{'}^2\}\; =\; \backslash frac\{k\text{'}^2\; -\; h\text{'}^2\}\{h\text{'}^2\}$; Attraction.

then, $F$ denoting the same fluent as before, the attractions of this ellipsoid upon the point within it, determined by the coordinates $a\text{'},\; b\text{'},\; c\text{'}$, in the directions of those coordinates, are (32.) respectively equal to

$$a\text{'}\; \backslash times\; \backslash frac\{3M\text{'}\}\{h^3\}\; \backslash left\backslash \{\; \backslash frac\{1\}\{\backslash sqrt\{(1+\backslash lambda^2)(1+\backslash lambda\text{'}^2)\}\}\; +\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right)\; +\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right)\; \backslash right\backslash \};$$ $$b\text{'}\; \backslash times\; \backslash frac\{3M\text{'}\}\{h\text{'}^3\}\; \backslash times\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right),$$ $$c\text{'}\; \backslash times\; \backslash frac\{3M\text{'}\}\{h\text{'}\text{'}^3\}\; \backslash times\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right).$$

Now, the attractions of the given ellipsoid upon the point without the surface, determined by the coordinates $a,\; b,\; c$, will be found (26.) by multiplying the preceding expressions respectively by $\backslash frac\{k\}\{h\},\; \backslash frac\{k\text{'}\}\{h\text{'}\},\; \backslash frac\{k\text{'}\text{'}\}\{h\text{'}\text{'}\}$,

Let $M$ be the mass of the given ellipsoid; then $\backslash frac\{M\}\{M\text{'}\}\; =\; \backslash frac\{k\; k\text{'}\; k\text{'}\text{'}\}\{h\; h\text{'}\; h\text{'}\text{'}\}$; consequently $\backslash frac\{k\; k\text{'}\}\{h\; h\text{'}\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash frac\{h\}\{k\}\; =\; \backslash frac\{M\}\{M\text{'}\}$.

$$\backslash frac\{a\}\{a\text{'}\}\; =\; \backslash frac\{h\}\{k\}\; \backslash times\; \backslash frac\{k\text{'}\}\{h\text{'}\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash times\; \backslash frac\{b\}\{b\text{'}\};\; \backslash text\{\; and\; \}\; \backslash frac\{k\; k\text{'}\}\{h\; h\text{'}\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash times\; \backslash frac\{c\}\{c\text{'}\};$$

$$\backslash frac\{a\}\{a\text{'}\}\; =\; \backslash frac\{h\}\{k\}\; \backslash times\; \backslash frac\{k\text{'}\}\{h\text{'}\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash times\; \backslash frac\{b\}\{b\text{'}\};\; \backslash text\{\; and\; \}\; \backslash frac\{k\; k\text{'}\}\{h\; h\text{'}\}\; =\; \backslash frac\{M\}\{M\text{'}\}\; \backslash times\; \backslash frac\{c\}\{c\text{'}\};$$

wherefore, the attractions of the given ellipsoid upon the point without the surface, determined by $a,\; b,\; c$, in the directions of those coordinates, are respectively equal

$$\backslash text\{to\; \}\; a\; \backslash times\; \backslash frac\{3M\}\{h^3\}\; \backslash times\; \backslash left\backslash \{\; \backslash frac\{1\}\{\backslash sqrt\{(1+\backslash lambda^2)(1+\backslash lambda\text{'}^2)\}\}\; +\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right)\; +\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right)\; \backslash right\backslash \},$$ $$b\; \backslash times\; \backslash frac\{3M\}\{h\text{'}^3\}\; \backslash times\; \backslash frac\{1\}\{\backslash lambda\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\}\; \backslash right)$$ $$c\; \backslash times\; \backslash frac\{3M\}\{h\text{'}\text{'}^3\}\; \backslash times\; \backslash frac\{1\}\{\backslash lambda\text{'}\}\; \backslash left(\; \backslash frac\{dF\}\{d\backslash lambda\text{'}\}\; \backslash right).$$

The preceding propositions contain a complete theory of homogeneous elliptical spheroids. They enable us to compute the attractive force with which a solid of this kind urges a particle placed anywhere in the surface, or within the solid, or without it. It remains, indeed, to find the exact value of the function $F$ in its general form, to which we can do no more than approximate by series; but this is an analytical difficulty, which it is impossible to overcome; because the nature of this function is such, that it cannot be expressed in finite terms by the received notation of analysis.

Attraction
||
Atwood.

In the preceding investigations, we have followed the method of Maclaurin for points situated in the surface of a spheroid, or within the solid. This method has always been justly admired; but neither its inventor, nor, as far as we know, any other Geometer, has applied it, except to spheroids of revolution; and it is here, for the first time, extended to ellipsoids. In regard to points without the surface, we have employed the method first given by Mr Ivory, in the Philosophical Transactions for 1809. The combination of these two methods has enabled us to derive the attractions of an ellipsoid on a point placed anywhere, from the attractions at the poles. Thus, this extremely complicated problem has, by geometrical reasoning of no great difficulty, been reduced to the investigation of the polar attractions, which are the only cases that require a direct computation.

34. Of the attractions of spheroids composed of elliptical shells that vary in their densities and figures according to any law. Attraction
||
Atwood.

When a spheroid is composed of concentric elliptical shells of variable density and figure, we may consider every shell as the difference of two homogeneous spheroids of the same density with the shells, and having their surfaces coinciding with the surfaces of the shell. The attractions of the spheroids being computed by the preceding methods, their difference will be equal to the attractions of the shell; and the integral obtained by summing the attractions of all the shells, will give the attractions of the heterogeneous spheroid. This case, therefore, gives rise to no new difficulties, except what are purely mathematical, and depend upon the law, according to which, the densities and figures of the shells are supposed to vary. (cc.)

ATTRACTION OF MOUNTAINS. See MOUNTAINS, ATTRACTION OF, in the Encyclopædia, and in this Supplement.

ATWOOD (GEORGE), an Author celebrated for the accuracy of his mathematical and mechanical investigations, and considered as particularly happy in the clearness of his explanations, and the elegance of his experimental illustrations, was born in the early part of the year 1746. He was educated at Westminster school, where he was admitted in 1759. Six years afterwards he was elected off to Trinity College, Cambridge. He took his degree of Bachelor of Arts in 1769, with the rank of third wrangler, Dr Parkinson, of Christ's College, being senior of the year. This distinction was amply sufficient to give him a claim to further advancement in his own College, on the list of which he stood foremost of his contemporaries; and, in due time, he obtained a fellowship, and was afterwards one of the tutors of the College. He became Master of Arts in 1772; and, in 1776, was elected a Fellow of the Royal Society of London.

The higher branches of the Mathematics had, at this period, been making some important advances at Cambridge, under the auspices of Dr Waring, and many of the younger members of the University became diligent labourers in this extensive field. Mr Atwood chose, for his peculiar department, the illustration of mechanical and experimental philosophy, by elementary investigations and ocular demonstrations of their fundamental truths. He delivered, for several successive years, a course of lectures in the Observatory of Trinity College, which were very generally attended, and greatly admired. In the year 1784, some circumstances occurred which made it desirable for him to discontinue his residence at Cambridge; and soon afterwards Mr Pitt, who had become acquainted with his merits by attending his lectures, bestowed on him a patent office, which required but little of his attendance, in order to have

a claim on the employment of his mathematical abilities in a variety of financial calculations, to which he continued to devote a considerable portion of his time and attention throughout the remainder of his life.*

The following, we believe, is a correct list of Mr Atwood's publications:

1. 1. A Description of Experiments to illustrate a Course of Lectures. 8vo. About 1775, or 1776.
2. 2. This work was reprinted with additions, under the title of An Analysis of a Course of Lectures on the Principles of Natural Philosophy. 8vo. Cambr. 1784.
3. 3. A General Theory for the Mensuration of the Angle subtended by two objects, of which one is observed by Rays after two Reflections from plane Surfaces, and the other by Rays coming directly to the Spectator's Eye. Phil. Trans. 1781, p. 395.
4. 4. A Treatise on the Rectilinear Motion and Rotation of Bodies, with a Description of Original Experiments relative to the Subject. 8vo. Cambr. 1784.
5. 5. Investigations founded on the Theory of Motion, for determining the Times of Vibration of Watch Balances. Phil. Trans. 1794, p. 119.
6. 6. The Construction and Analysis of Geometrical Propositions, determining the positions assumed by homogeneous bodies, which float freely, and at rest, on a fluid's surface; also Determining the Stability of Ships, and of other Floating Bodies. Phil. Trans. 1796, p. 46.
7. 7. A Disquisition on the Stability of Ships. Phil. Trans. 1798, p. 201.
8. 8. A Review of the Statutes and Ordinances of Assize, which have been established in England from the 4th year of King John, 1202, to the 37th of his present Majesty. 4to. Lond. 1801.
9. 9. A Dissertation on the Construction and Properties of Arches. 4to. Lond. 1801.
10. 10. A Supplement to a Tract entitled a Treatise on the Construction and Properties of Arches, published

* See Literary Memoirs of Living Authors, 2 vol. 8vo, Lond. 1798; Morning Herald, 17th July 1807; Nichols's Literary Anecdotes of the Eighteenth Century, Vol. VIII. 8vo, Lond. 1814.

Fig. 1.

A geometric diagram showing a circular segment with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The segment is bounded by points A and B, with points C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z distributed along the arc and the chord.

Fig. 2.

A geometric diagram showing two concentric circles with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of both circles and the radii connecting their centers.

A perspective view of a rectangular prism with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The prism is shown in perspective, with lines connecting the vertices to a vanishing point.

Fig. 4.

A perspective view of a rectangular prism with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The prism is shown in perspective, with lines connecting the vertices to a vanishing point.

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A geometric diagram showing two concentric circles with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of both circles and the radii connecting their centers.

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A geometric diagram showing two concentric circles with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of both circles and the radii connecting their centers.

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A perspective view of a rectangular prism with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The prism is shown in perspective, with lines connecting the vertices to a vanishing point.

Fig. 7.

A geometric diagram showing a circle with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of the circle and the radii connecting their centers.

Fig. 7.

A geometric diagram showing a circle with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of the circle and the radii connecting their centers.

Fig. 8.

A geometric diagram showing two concentric circles with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of both circles and the radii connecting their centers.

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A geometric diagram showing a circle with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of the circle and the radii connecting their centers.

Fig. 10.

A geometric diagram showing a circle with points A, B, C, D, E, F, G, H, I, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z. The points are distributed along the circumference of the circle and the radii connecting their centers.

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This image shows a blank, aged, cream-colored page, likely an endpaper or flyleaf from an old book. The paper has a slightly textured appearance with some minor discoloration and faint, repeating circular patterns that suggest a watermark or a decorative design. There are also some small, dark spots or stains scattered across the surface. The right edge of the page shows a slight shadow, indicating it is part of a bound volume.