FLUXIONS.

method of calculation which greatly facilitates computations in the higher parts of mathematics. Sir Isaac Newton and Mr Leibnitz contended for the honour of inventing it. It is probable they had both made progress in the same discovery, unknown to each other, before there was any publication on the subject.

In this branch of mathematics magnitudes of every kind are supposed generated by motion: a line by the motion of a point, a surface by the motion of a line, and a solid by the motion of a surface. And some part of a figure is supposed generated by an uniform motion; in consequence of which the other parts may increase uniformly or with an accelerated or retarded motion, or may decrease in any of these ways; and the computations are made by tracing the comparative velocities with which the parts flow.

Fig. 1. If the parallelogram ABCD be generated by an uniform motion of the line AB toward CD while it moves from FE towards fE, while the line BF receives the increment Ff, and the figure will be increased by the parallelogram Fe; the line FE in this case undergoes no variation.

The fluxion of any magnitude at any point is the increment that it would receive in any given time, supposing it to increase uniformly from that point; and as the measures will be the same, whatever the time be, we are at liberty to suppose it less than any assigned time.

The first letters in the alphabet are used to represent invariable quantities; the letters x, y, z variable quantities; and the same letters with points over them \(x\), \(y\), \(z\), represent their fluxions.

Therefore if \(AB = a\), and \(BF = x\); \(Ff\), the fluxion of \(BF\), will be \(= x\), and \(Fe\), the fluxion of \(AF\), \(= ax\).

If the rectangle be supposed generated by the uniform motion of FG towards CD, at the same time HG moves uniformly towards AD, the point G keeping always on the diagonal; the lines FG HG will flow uniformly; for while \(BF\) receives the increment \(Ff\) and HB, the increment HK, FG will receive the increment \(hg\) and HG the increment \(bg\), and they will receive equal increments in equal successive times. But the parallelogram will flow with an accelerated motion; for while F flows to \(f\) and H to K, it is increased by the gnomon KGf; but while F and H flow through the equal spaces \(fm\) KL, it is increased by the gnomon Lgm greater than KGf; consequently when fluxions of the sides of the parallelogram are uniform, the fluxion of the parallelogram increases continually.

The fluxion of the parallelogram BHGF is the two parallelograms KG and Gf; for though the parameter receives an increment of the gnomon KGf, while its sides flow to \(f\) and K, the part \(gG\) is owing to the additional velocity wherewith the parallelogram flows during that time; and therefore is no part of the measure of the fluxion, which must be computed by supposing the parameter to flow uniformly as it did at the beginning, without any acceleration.

Therefore if the sides of a parallelogram be \(x\) and \(y\), their fluxions will be \(x\) \(y\); and the fluxion of the parallelogram \(xy + yx\); and if \(x = y\), that is, if the figure be a square, the fluxion of \(x^2\) will be \(2xx\).

Fig. 2. Let the triangle ABC be described by the uniform motion of DE from A towards B, the point E moving in the line DF, so as always to touch the lines AC, CB; while D moves from A to F, DE is uniformly increased, and the increase of the triangle is uniformly accelerated. When DE is in the position FC, it is a maximum. As D moves from F to B, the line FC decreases, and the triangle increases, but with a motion uniformly retarded.

Fig. 3. If the semicircle AFB be generated by the uniform motion of CD from A towards B, while C moves from A to G, the line CD will increase, but with a retarded motion; the circumference also increases with a retarded motion, and the circular space increases with an accelerated motion, but not uniformly, the degrees of acceleration growing less as CD approaches to the position GF. When C moves from G to B, it decreases with a motion continually accelerated, the circumference increases with a motion continually accelerated, and the area increases with a motion continually retarded, and more quickly retarded as CD approaches to B.

The fluxion of a quantity which decreases is to be considered as negative.

When a quantity does not flow uniformly, its fluxion may be represented by a variable quantity, or a line of a variable length; the fluxion of such a line is called the second fluxion of the quantity whose fluxion that line is; and if it be variable, a third fluxion may be deduced from it, and higher orders from these in the same manner; the second fluxion is represented by two points, as \(x\).

The increment a quantity receives by flowing for any given time, contains measures of all the different orders of fluxions; for if it increases uniformly, the whole increment is the first fluxion; and it has no second fluxion. If it increases with a motion uniformly accelerated, the part of the increment occasioned by the first motion measures the first fluxion, and the part occasioned by the acceleration measures the second fluxion. If the motion be not only accelerated, but the degree of acceleration continually increased, the two first fluxions are measured as before; and the part of the increment occasioned by the additional degree of acceleration measures the third; and so on. These measures require to be corrected, and are only mentioned here to illustrate the subject.

DIRECT METHOD.

Any flowing quantity being given, to find its fluxion.

Rule I. To find the fluxion of any power of a quantity

Multiply the fluxion of the root by the exponent of the power, and the product by a power of the same root less by unity than the given exponent.

The fluxion of \( x^n \) is \( nx^{n-1}x \); for the root of \( x^n \) is \( x \), whose fluxion is \( x \); which multiplied by the exponent \( n \), and by a power of \( x \) less by unity than \( n \), gives the above fluxion.

If \( x \) receive the increment \( \dot{x} \), it becomes \( x + \dot{x} \); raise both to the power of \( n \), and \( x^n \) becomes \( x^n + nx^{n-1}\dot{x} + \frac{n(n-1)}{2}x^{n-2}\dot{x}^2 + \ldots \); but all the parts of the increment, except the first term, are owing to the accelerated increase of \( x^n \), and form measures of the higher fluxions. The first term only measures the first fluxion; the fluxion of \( a^3 + z^2 \) is \( \frac{3}{2}a^2z + az^2 \); for put \( x = a^3 + z^2 \), we have \( x = 2zz \), and the fluxion of \( x^3 \), which is equal to the proposed fluent, is \( \frac{3}{2}x^2 \), for which substituting the values of \( x \) and \( z \), we have the above fluxion.

Rule II. To find the fluxion of the product of several variable quantities multiplied together, multiply the fluxion of each by the product of the rest of the quantities, and the sum of the products thus arising will be the fluxion sought.

Thus the fluxion of \( xy \), is \( y\dot{x} + x\dot{y} \); that of \( xyz \), is \( xyz\dot{y} + xzy\dot{z} + yzx\dot{z} \); and that of \( xyzu \), is \( xyzu\dot{z} + xzyu\dot{z} + yzxu\dot{z} + yzux\dot{z} \).

Rule III. To find the fluxion of a fraction.—From the fluxion of the numerator multiplied by the denominator, subtract the fluxion of the denominator multiplied by the numerator, and divide the remainder by the square of the denominator.

Thus, the fluxion of \( \frac{x}{y} \) is \( \frac{y\dot{x} - x\dot{y}}{y^2} \); that of \( \frac{x}{x+y} \) is \( \frac{x(x+y)\dot{y} - (x+y)x\dot{y}}{(x+y)^2} = \frac{y\dot{x} - x\dot{y}}{x+y} \).

Rule IV. In complex cases, let the particulars be collected from the simple rules and combined together.

The fluxion of \( \frac{x^2y^2}{z} \) is \( \frac{2x^2y\dot{y} + 2y^2xx\dot{x} - x^2y^2\dot{z}}{z^2} \); for the fluxion of \( x^2 \) is \( 2xx \), and of \( y^2 \) is \( 2yy \), by Rule I., and therefore the fluxion of \( x^2y^2 \) (by Rule II.) \( 2x^2y\dot{y} + 2y^2xx\dot{x} \); from which, multiplied by \( z \), (by Rule III.), and subtracting from it the fluxion of the denominator \( z \), multiplied by the numerator, and dividing the whole by the square of the denominator, gives the above fluxion.

Rule IV. The second fluxion is derived from the first, in the same manner as the first from the flowing quantity.

Thus the fluxion of \( x^3 \), \( 3x^2\dot{x} \); its second, \( 6x\dot{x}^2 + 3x^2\dot{x} \) (by Rule II.); and so on: but if \( x \) be invariable, \( x = 0 \), and the second fluxion of \( x^3 = 6x\dot{x}^2 \).

Prob. 1. To determine maxima and minima

When a quantity increases, its fluxion is positive; when it decreases, it is negative; therefore when it is just betwixt increasing and decreasing, its fluxion is \( = 0 \).

Rule. Find the fluxion, make it \( = 0 \), whence an equation will result that will give an answer to the question.

Fig. 4. Example. To determine the dimensions of a cylindric measure ABCD, open at the top, which shall contain a given quantity (of liquor, grain, &c.) under the least internal superficies possible.

Let the diameter \( AB = x \), and the altitude \( AD = y \); moreover, let \( p \) (3.14159, &c.) denote the periphery of the circle whose diameter is unity, and let \( c \) be the given content of the cylinder. Then it will be \( 1 : p :: x : (px) \) the circumference of the base; which, multiplied by the altitude \( y \), gives \( pxy \) for the concave superficies of the cylinder. In like manner, the area of the base, by multiplying the same expression into \( \frac{1}{4} \) of the diameter \( x \), will be found \( = \frac{px^2}{4} \); which drawn into the altitude \( y \), gives \( \frac{pxy}{4} \) for the solid content of the cylinder; which being made \( = c \), the concave surface \( pxy \) will be found \( = \frac{4c}{x} \) and consequently the whole surface \( = \frac{4c}{x} + \frac{px^2}{4} \): Whereof the fluxion, which is \( -\frac{4c}{x^2} + \frac{px}{2} \) being put \( = 0 \), we shall get \( -8c + px^2 = 0 \); and therefore \( x = 2\sqrt{\frac{c}{p}} \): further, because \( px^2 = 8c \), and \( px^2 = 4c \), it follows, that \( x = 2y \); whence \( y \) is also known, and from which it appears, that the diameter of the base must be just the double of the altitude.

Fig. 7. To find the longest and shortest ordinates of any curve, DEF, whose equation or the relation which the ordinates bear to the abscissas is known.

Make AC the abscissa \( x \), and CE the ordinate \( y \); take a value \( y \) in terms of \( x \), and find its fluxion; which making \( = 0 \), an equation will result whose roots give the value of \( x \) when \( y \) is a maximum or minimum.

To determine when it is a maximum and when a minimum, take the value of \( y \), when \( x \) is a little more than the root of the equation so found, and it may be perceived whether it increases or decreases.

If the equation has an even number of equal roots, \( y \) will be neither a maximum nor minimum when its fluxion is \( = 0 \).

Prob. 2. To draw a tangent to any curve.

Fig. 5. When the abscissa CS of a curve moves uniformly from A to B, the motion of the curve will be retarded if it be concave, and accelerated if convex towards AB; for a straight line TC is described by an uniform motion, and the fluxion of the curve at any point is the same as the fluxion of the tangent, because it would describe the tangent if it continued to move equally from that point. Now if S, or Ce be the fluxion of the base, Cd will be the fluxion of the tangent, and de of the ordinate. And because the triangles TSC, ced, are equiangular, \( de : ce :: CS : ST \), wherefore

Rule. Find a fourth proportional to the fluxion of the ordinate valued in terms of the abscissa, the fluxion of the abscissa, and the ordinate, and it determines the line ST, which is called the semi-tangent, and TC joined is a tangent to the curve.

Fig. 6. Examp. To draw a right line CT, to touch a given circle BCA in a given point C.

Let CS be perpendicular to the diameter AB, and put \( AB = a \), \( BS = x \), and \( SC = y \): Then, by the property of the circle, \( y^2 (CS^2) = BS \times AS \) \( = ax - x^2 \); whereof the fluxion being taken, in order to determine the ratio of \( \dot{x} \) and \( \dot{y} \), we get \( 2yy = ax - 2xx \); consequently

\[ \frac{\dot{x}}{y} = \frac{2y}{a - 2x} = \frac{y}{\frac{a}{2} - x} \]

which multiplied by \( y \), gives \( \frac{yx}{y} = \frac{y^2}{\frac{a}{2} - x} \)

\( = \frac{y^2}{\frac{a}{2} - x} \) = the subtangent ST. Whence (O being supposed the centre) we have \( OS (\frac{a}{2} - x) : CS (y) :: CS (y) : ST \); which we also know from other principles.

Prob. 3. To determine points of contrary flexure in curves.

Fig. 7. Supposing C to move uniformly from A to B, the curve DEF will be convex towards AB when the celerity of E increases, and concave when it decreases; therefore at the point where it ceases to be convex and begins to be concave, or the opposite way, the celerity of E will be uniform, that is, CE will have no second fluxion.

Therefore,

Rule. Find the second fluxion of the ordinate in terms of the abscissa, and make it = 0; and from the equation that arises you get a value of the abscissa, which determines the point of contrary flexure.

Ex. Let the nature of the curve ARS be defined by the equation \( ay = \frac{1}{4}x^{\frac{3}{2}} + xx \), (the abscissa AF and the ordinate FG being, as usual, represented by \( x \) and \( y \) respectively). Then \( y \), expressing the celerity of the point \( r \),

in the line FH, will be equal to \( \frac{1}{4}a^{\frac{3}{2}}x^{\frac{1}{2}} + 2xx \): Whose fluxion, or that of \( \frac{1}{4}a^{\frac{3}{2}}x^{\frac{1}{2}} + 2x \) (because \( a \) and \( x \) are constant) must be equal to nothing; that is, \( -\frac{1}{4}a^{\frac{3}{2}}x^{-\frac{1}{2}} + 2x = 0 \): Whence \( a^{\frac{3}{2}}x^{-\frac{1}{2}} = 8, a^{\frac{3}{2}} = 8x^{\frac{3}{2}}, 64x^{\frac{3}{2}} = a^{\frac{3}{2}} \), and \( x = \frac{a}{4} = AF \); therefore \( FG = \frac{a^{\frac{3}{2}}x^{\frac{1}{2}} + 2x}{a} = \frac{a}{4} \): From which the position of the point G is given.

Prob. 4. To find the radii of curvature.

The curvature of a circle is uniform in every point, that of every other curve continually varying; and it is measured at any point by that of a circle whose radius is of such a length as to coincide with it in curvature in that point.

All curves that have the same tangent have the same first fluxion, because the fluxion of a curve and its tangent are the same. If it moved uniformly on from the point of contact, it would describe the tangent. And the deflection from the tangent is owing to the acceleration or retardation of its motion, which is measured by its second fluxion; and consequently two curves which have not only the same tangent, but the same curvature at the point of contact, will have both their first and second fluxions equal. It is easily proven from thence, that the radius of curvature is \( \frac{z^3}{xy} \), where \( x, y, \) and \( z \) represent the abscissa, ordinate, and curve respectively.

Examp. Let the given curve be the common parabola, whose equation is \( y = a^{\frac{1}{2}}x^{\frac{1}{2}} \): Then will \( y = \frac{1}{2}a^{\frac{1}{2}}x^{\frac{1}{2}} \)

\( = \frac{a^{\frac{1}{2}}x}{2x^{\frac{1}{2}}} \), and (making \( x \) constant) \( y = \frac{1}{2}a^{\frac{1}{2}}x^{\frac{1}{2}} \)

\( = \frac{a^{\frac{1}{2}}x^{\frac{1}{2}}}{4x^{\frac{1}{2}}} \): Whence \( z (\sqrt{x^2 + y^2}) = \frac{x}{2} \sqrt{\frac{4x + a}{x}} \), and the radius of curvature \( \frac{z^3}{xy} = \frac{a + 4x^{\frac{3}{2}}}{24x^{\frac{3}{2}}} \): Which at the vertex, where \( x = 0 \), will be \( \frac{a}{2} \).

Inverse Method:

From a given fluxion to find a fluent.

This is done by tracing back the steps of the direct method. The fluxion of \( x \) is \( \dot{x} \); and therefore the fluent of \( x \) is \( x \); but as there is no direct method of finding fluents, this branch of the art is imperfect. We can assign the fluxion of every fluent, but we cannot assign the fluent of a fluxion, unless it be such a one as may be produced by some rule in the direct method from a known fluent.

General Rule. Divide by the fluxion of the root, add unity to the exponent of the power, and divide by the exponent so increased.

For, dividing the fluxion \( nx^{n-1}\dot{x} \) by \( \dot{x} \) (the fluxion of the root \( x \)) it becomes \( nx^{n-1} \); and, adding 1 to the exponent \( (n-1) \) we have \( nx^n \); which, divided by \( n \), gives \( x^n \), the true fluent of \( nx^{n-1}\dot{x} \).

Hence (by the same rule) the

Fluent of \( 3x^2 \dot{x} \) will be \( = x^3 \);

That of \( 8x^3 \dot{x} \) \( = \frac{8x^3}{3} \);

That of \( 2x^3 \dot{x} \) \( = \frac{x^6}{3} \);

That of \( y^2 \dot{y} = \frac{y^3}{3} \).

Sometimes the fluent so found requires to be corrected. The fluxion of \( x \) is \( \dot{x} \), and the fluxion of \( a + x \) is also \( \dot{x} \), because \( a \) is invariable, and has therefore no fluxion.

Now when the fluent of \( x \) is required, it must be determined, from the nature of the problem, whether any invariable part, as \( a \), must be added to the variable part \( x \).

When fluents cannot be exactly found, they can be approximated by infinite series.

Ex. Let it be required to approximate the fluent of

\[ \frac{a^{\frac{1}{2}} - x^{\frac{1}{2}}}{c^{\frac{1}{2}} - x^{\frac{1}{2}}} \cdot \frac{1}{x} \cdot \frac{1}{x} \text{ in an infinite series.} \] The value of \( \frac{a^2 - x^2}{c^2 - x^2} \), expressed in a series, is \( \frac{a}{c} + \frac{a}{2c^3 - 2ac} \times x^2 + \frac{3a}{8c^5 - 4ac^3} \times x^4 + \frac{5a}{16c^7 - 16ac^5} \times x^6 + \ldots \). Which value being therefore multiplied by \( x^n \), and the fluent taken (by the common method) we get \( \frac{ax^{n+1}}{n+1} + \frac{a}{2c^3 - 2ac} \times \frac{x^{n+3}}{n+3} + \frac{3a}{8c^5 - 4ac^3} \times \frac{x^{n+5}}{n+5} + \ldots \).

**Prob. 1.** To find the area of any curve.

**Rule.** Multiply the ordinate by the fluxion of the abscissa, and the product gives the fluxion of the figure, whose fluent is the area of the figure.

**Examp. 1.** Fig. 8. Let the curve ARMH, whose area you will find, be the common parabola. Let \( u \) represent the area, and \( u \) its fluxion.

In which case the relation of \( AB(x) \) and \( BR(y) \) being expressed by \( y = ax \) (where \( a \) is the parameter) we hence get \( y = u^{\frac{1}{2}}x^{\frac{1}{2}} \); and therefore \( u = RmHB (= y^2) = \frac{1}{3}x^3 \); whence \( u = \frac{2}{3}x^{\frac{3}{2}} \times x^{\frac{1}{2}}y \) (because \( a^{\frac{1}{2}}x^{\frac{1}{2}} = y \)) \( = \frac{2}{3} \times AB \times BR \); hence a parabola is \( \frac{2}{3} \) of a rectangle of the same base and altitude.

**Examp. 2.** Let the proposed curve CSDR (fig. 9.) be of such a nature, that (supposing AB unity) the sum of the areas CSTBC and CDGBC answering to any two proposed abscissas AT and AG, shall be equal to the area CRNB, whose corresponding abscissa AN is equal to \( AT \times AG \), the product of the measures of the two former abscissas.

First, in order to determine the equation of the curve, (which must be known before the area can be found) let the ordinates GD and NR move parallel to themselves towards HF; and then having put \( GD = y \), \( NR = z \), \( AT = s \), \( AG = r \), and \( AN = u \), the fluxion of the area CDGB will be represented by \( y_i \), and that of the area CRNB by \( zu \): which two expressions must, by the nature of the problem, be equal to each other; because the latter area CRNB exceeds the former CDGB by the area CSTB, which is here considered as a constant quantity: and it is evident, that two expressions, that differ only by a constant quantity, must always have equal flexions.

Since, therefore, \( y_i \) is \( = zu \), and \( u = ar \), by hypothesis, it follows, that \( u = ar \), and that the first equation (by substitution for \( u \)) will become \( y_i = az \), or \( v = az \), or lastly \( y_i = za \); that is, \( GD \times AG = NR \times AN \): therefore, \( GD : NR :: AN : AG \); whence it appears, that every ordinate of the curve is reciprocally as its corresponding abscissa.

Now, to find the area of the curve so determined, put \( AB = 1 \), \( BC = b \), and \( BC = x \); then, since \( AG(1+x) : AB(1) :: BC(b) : GD(y) \) we have \( y = \frac{b}{1+x} \), and consequently \( u = \frac{bx}{1+x} = b \times x - xx + x^2x - x^3x + x^4x + \ldots \). Whence, \( BGDC \), the area itself will be \( b \times x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} + \ldots \), which was to be found.

Hence it appears, that as these areas have the same properties as logarithms, this series gives an easy method of computing logarithms; and the fluent may be found by means of a table of logarithms, without the trouble of an infinite series: and every fluxion whose fluent agrees with any known logarithmic expression, may be found the same way. Hence the fluents of fluxions of the following forms are deduced.

The fluent of \( \frac{x}{\sqrt{x^2 + a^2}} \) = hyp. log. of \( x + \sqrt{x^2 + a^2} \);

of \( \frac{x}{\sqrt{2ax + xx}} \) = hyp. log. \( ax + \sqrt{2ax + x^2} \);

of \( \frac{2ax}{a^2 - x^2} \) = hyp. log. of \( \frac{a + x}{a - x} \);

and of \( \frac{2ax}{x\sqrt{a^2 - x^2}} \) = hyp. log. \( \frac{a - \sqrt{a^2 - x^2}}{a + \sqrt{a^2 - x^2}} \).

**Prob. 2.** To determine the length of curves.

Fig. 5. Because Cde is a right-angled triangle, \( Cd^2 = Ce^2 + de^2 \); wherefore the fluxions of the abscissa and ordinate being taken in the same terms and squared, their sum gives the square of the fluxion of the curve; whose root being extracted, and the fluent taken, gives the length of the curve.

**Examp.** To find the length of a circle from its tangent. Make the radius AO (fig. 5.) \( = a \), the tangent of AC \( = t \), and its secant \( = s \), the curve \( = z \), and its fluxion \( = z_i \); because the triangles OTC, OCS, are similar, \( OT : OC :: OC : OS \); whence \( OS = \frac{a^2}{s} \), and \( SA = a - \frac{a^2}{s} = a - \frac{a^2}{\sqrt{a^2 + t^2}} \); whose fluxion is \( \frac{a^2it}{a^2 + t^2} \); and because the triangles OTC, dCe are similar, \( TC = \frac{a^2i}{a^2 + t^2} \); \( TO = \frac{a^2i}{a^2 + t^2} \); \( C d = \frac{a^2i}{a^2 + t^2} \); fluxion of the curve.

Now by converting this into an infinite series, we have the fluxion of the curve \( = \frac{t^2i}{a^2} + \frac{t^4i}{a^4} - \frac{t^6i}{a^6} + \ldots \), and consequently \( z = \frac{t^3}{3a^2} + \frac{t^5}{5a^4} - \frac{t^7}{7a^6} + \frac{t^9}{9a^8} + \ldots \), \( AR \).

Where, if (for example's sake) AR be supposed an arch of 30 degrees, and AO (to render the operation more easy) be put = unity, we shall have \( t = \sqrt{\frac{1}{4}} = 0.5773502 \) (because \( Ob/\sqrt{4} : bR(\frac{1}{2}) :: OA(1) : AT(t) = \sqrt{\frac{1}{4}} \)).

Whence,

\( \frac{t^3}{3} = (\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4}) = 0.1924500 \)

\( \frac{t^5}{5} = (\frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4}) = 0.0641500 \)

\[ \begin{align*} t^7 &= \frac{t^5 \times t^2}{3} = .0213833 \\ t^9 &= \frac{t^7 \times t^2}{3} = .0071277 \\ t^{11} &= \frac{t^9 \times t^2}{3} = .0023759 \\ t^{13} &= \frac{t^{11} \times t^2}{3} = .0007919 \\ t^{15} &= \frac{t^{13} \times t^2}{3} = .0002639 \\ &\text{etc.} \end{align*} \]

And therefore \( AR = .5773502 - \frac{1.934500}{3} + \frac{.0641500}{5} \times \frac{.0213833}{7} \times \frac{.0091277}{9} \times \frac{.0023759}{11} + \frac{.0007919}{13} \times \frac{.0002639}{15} \times \frac{.0000879}{17} \times \frac{.0000293}{19} + \frac{.0000097}{21} \times \frac{.0000032}{23} = .5235987; \) for the length of an arch of 30 degrees, which multiplied by 6 gives 3.141592 + for the length of the semi-periphery of the circle whose radius is unity.

Other series may be deduced from the versed sine, fine and secant; and these are of use for finding fluents which cannot be expressed in finite terms. For,

\[ \begin{align*} \text{Versed-sine} &= \sqrt{2aw - w^2} \\ \text{Right-sine} &= \sqrt{a^2 - w^2} \\ \text{Tangent} &= \frac{aw}{a^2 + w^2} \\ \text{Secant} &= \frac{aw}{w\sqrt{w^2 - a^2}} \end{align*} \]

is equal to the arch whose

\[ \text{Radius Unity.} \]

PROB. 3. To find the contents of a solid.

Let the surface of the generating plane be multiplied by the space it passes through in any time, the product will give a solid which is the fluxion of the solid required; the surface must therefore be computed in terms of \( x \), which represents the line or axis on which it moves, and by its motion on which the fluxion is to be measured, and the fluent found will give the contents of the solid.

FLY