FLUXIONS

FLUXIONS, a method of calculation which greatly facilitates computations in the higher parts of mathematics. Sir Isaac Newton and Mr Leibnitz contend 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 $\backslash dot\{x\},\; \backslash dot\{y\},\; \backslash dot\{z\}$, represent their fluxions.

Therefore if $AB\; =\; a$, and $BF\; =\; x$; $Ff$, the fluxion of $BF$, will be $\backslash dot\{x\}$, and $Fe$, the fluxion of $AF$, $=\; a\backslash dot\{x\}$.

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 $hg$, 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 a 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 para-

meter 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 $\backslash dot\{x\}$ $\backslash dot\{y\}$; and the fluxion of the parallelogram $\backslash dot\{x\}y\; +\; x\backslash dot\{y\}$; 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 $\backslash ddot\{x\}$.

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.

Any flowing quantity being given, to find its fluxion.

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

ity, 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 $nx^\{n-1\}x$, of $x^n\; \backslash cdot\; nx^\{n-1\}x$; 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 $x$, it becomes $x+x$; raise both to the power of $n$, and $x^n$ becomes $x^n\; +\; nx^\{n-1\}x\; +\; \backslash frac\{n(n-1)\}\{2\}x^\{n-2\}x^2\; +\; \backslash dots$; 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 $a^2+z^2$ is $\backslash frac\{1\}\{2\}\; \backslash times\; 2zz\; \backslash times\; a^2\; +\; z^2$; for put $x=a^2+z^2$, we have $x=2zz$, and the fluxion of $x^\{\backslash frac\{1\}\{2\}\}$, which is equal to the proposed fluent, is $\backslash frac\{1\}\{2\}x^\{\backslash frac\{1\}\{2\}\}x$, for which substituting the values of $x$ and $x$, 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 $xy+yx$; that of $xyz$, is $xyz+xyzy+yzx$; and that of $xyzu$, is $xyzu+xyuz+xyzy+yzxu$.

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 $\backslash frac\{x\}\{y\}$ is $\backslash frac\{yx-xy\}\{y^2\}$; that of $\backslash frac\{x\}\{x+y\}$, is $\backslash frac\{x\; \backslash times\; x+y\; -\; x+y\; \backslash times\; x\}\{(x+y)^2\}\; =\; \backslash frac\{yx-xy\}\{(x+y)^2\}$.

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

The fluxion of $\backslash frac\{x^2y^2\}\{z\}$ is $\backslash frac\{2x^2yy+2y^2xx\; \backslash times\; z\; -\; x^2y^2z\}\{z^2\}$; 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) $2x^2yy+2y^2xx$; 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 $x^3$, $3x^2x$; its second $6xx^2+3x^2x$ (by Rule II); and so on: but if $x$ be invariable, $x=0$, and the second fluxion of $x^3=6x^2$.

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 between 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. EXAMP. 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 $\backslash frac\{1\}\{2\}$ of the diameter $x$, will be found $=\backslash frac\{px^2\}\{4\}$; which drawn into the altitude $y$, gives

$\backslash frac\{px^2y\}\{4\}$ for the solid content of the cylinder; which being

made $=c$, the concave surface $pxy$ will be found $=\backslash frac\{4c\}\{x\}$

and consequently the whole surface $=\backslash frac\{4c\}\{x\}\; +\; \backslash frac\{px^2\}\{4\}$: Where-

of the fluxion, which is $-\backslash frac\{4c\}\{x^2\}\; +\; \backslash frac\{px\}\{2\}$ being put $=0$, we

shall get $-8c\; +\; px^3\; =\; 0$; and therefore $x\; =\; 2\backslash sqrt[3]\{\backslash frac\{c\}\{p\}\}$: further,

because $px^3=8c$, and $px^2y=4c$, 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, $de\; :\; c\; ::\; CS\; :\; ST$, wherefore

RULE. Find a fourth proportional to the fluxion of the

Fig. 1

FLUXIONS

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, $y^2(CS^2)=BS\; \backslash times\; AS\; (=x\; \backslash times\; a-x)=ax-x^2$; whereof the fluxion being taken, in order to determine the ratio of $\backslash dot\{x\}$ and $\backslash dot\{y\}$, we get $2yy=2ax-2x^2$; consequently $\backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}=\backslash frac\{2y\}\{a-2x\}=\backslash frac\{y\}\{\backslash frac\{1\}\{2\}a-x\}$; which multiplied by $\backslash dot\{y\}$, gives $\backslash frac\{y\backslash dot\{x\}\}\{\backslash frac\{1\}\{2\}a-x\}=\backslash frac\{y\backslash dot\{x\}\}\{\backslash frac\{1\}\{2\}a-x\}$ = the subtangent ST. Whence (O being supposed the centre) we have $OS\; (\backslash frac\{1\}\{2\}a-x)\; :\; CS\; (y)\; ::\; CS\; (y)\; :\; ST$; 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 $ay=a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}+xx$, (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 $\backslash frac\{\backslash frac\{1\}\{2\}ax^\{\backslash frac\{1\}\{2\}\}+2xx\}\{a\}$: Whose fluxion, or that of $\backslash frac\{1\}\{2\}a^\{\backslash frac\{1\}\{2\}\}x^\{-\backslash frac\{1\}\{2\}\}+2x$ (because $a$ and $\backslash dot\{x\}$ are constant) must be equal to nothing; that is, $-\backslash frac\{1\}\{4\}a^\{\backslash frac\{1\}\{2\}\}x^\{-\backslash frac\{3\}\{2\}\}+2x=0$: Whence $a^\{\backslash frac\{1\}\{2\}\}x^\{-\backslash frac\{3\}\{2\}\}=8$, $a^\{\backslash frac\{1\}\{2\}\}=8x^\{\backslash frac\{3\}\{2\}\}$, $64x^3=a^2$, and $x=\backslash frac\{1\}\{2\}a=AF$; therefore $FG\; (=a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}+xx)=\backslash frac\{1\}\{2\}a$: 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 $=\backslash frac\{z^3\}\{-xy\}$, where $x$, $y$, and $z$ represent the abscissa, ordinate, and curve respectively.

EXAMP. Let the given curve be the common parabola, whose equation is $y=a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}$: Then will $\backslash dot\{y\}=\backslash frac\{1\}\{2\}a^\{\backslash frac\{1\}\{2\}\}x^\{-\backslash frac\{1\}\{2\}\}$, $=\backslash frac\{a^\{\backslash frac\{1\}\{2\}\}\backslash dot\{x\}\}\{2x^\{\backslash frac\{1\}\{2\}\}\}$, and (making $\backslash dot\{x\}$ constant) $\backslash ddot\{y\}=-\backslash frac\{1\}\{4\}a^\{\backslash frac\{1\}\{2\}\}x^\{-\backslash frac\{3\}\{2\}\}$: Whence $z(\backslash sqrt\{x^2+y^2\})=\backslash frac\{\backslash dot\{x\}\}\{2\}\; \backslash sqrt\{\backslash frac\{4x+a\}\{x\}\}$, and

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

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 $\backslash dot\{x\}$; and therefore the fluent of $\backslash dot\{x\}$ 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 $nx^\{n-1\}\backslash dot\{x\}$ by $\backslash dot\{x\}$ (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 $nx^\{n-1\}\backslash dot\{x\}$.

Hence (by the same rule) the

Fluent of $3x^2\backslash dot\{x\}$ will be $=x^3$;

That of $8x^2\backslash dot\{x\}=\backslash frac\{8x^3\}\{3\}$;

That of $2x^2\backslash dot\{x\}=\backslash frac\{x^3\}\{3\}$;

That of $y^\{\backslash frac\{1\}\{2\}\}\backslash dot\{y\}=\backslash frac\{1\}\{2\}y^\{\backslash frac\{3\}\{2\}\}$.

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

Now when the fluent of $\backslash dot\{x\}$ 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

The value of $\backslash frac\{\backslash sqrt\{a^2-x^2\}\}\{c^2-x^2\}$, expressed in a series, is $\backslash frac\{a\}\{c\}\; +\; \backslash frac\{a\}\{2c^3\}\; -\; \backslash frac\{1\}\{2ac\}\; \backslash times\; x^2\; +\; \backslash frac\{3a\}\{8c^5\}\; -\; \backslash frac\{1\}\{4ac^3\}\; -\; \backslash frac\{1\}\{8a^3c\}\; \backslash times\; x^4\; +\; \backslash frac\{5a\}\{16c^7\}\; -\; \backslash frac\{3\}\{16ac\}\; -\; \backslash frac\{1\}\{16a^3c^3\}\; -\; \backslash frac\{1\}\{16a^5c\}\; \backslash times\; x^6\; +\; \backslash dots$. Which value being therefore multiplied by $x^n\; x$, and the fluent taken (by the common method) we get $\backslash frac\{ax^\{n+1\}\}\{n+1\}\; +\; \backslash frac\{a\}\{2c^3\}\; -\; \backslash frac\{1\}\{2ac\}\; \backslash times\; \backslash frac\{x^\{n+3\}\}\{n+3\}\; +\; \backslash frac\{3a\}\{8c^5\}\; -\; \backslash frac\{1\}\{4ac^3\}\; -\; \backslash frac\{1\}\{8a^3c\}\; \backslash times\; \backslash frac\{x^\{n+5\}\}\{n+5\}\; +\; \backslash frac\{5a\}\{16c^7\}\; -\; \backslash frac\{3\}\{16ac\}\; -\; \backslash frac\{1\}\{16a^3c^3\}\; -\; \backslash frac\{1\}\{16a^5c\}\; \backslash times\; \backslash frac\{x^\{n+7\}\}\{n+7\}\; +\; \backslash dots$

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 $\backslash dot\{u\}$ its fluxion.

In which case the relation of AB ($x$) and BR ($y$) being expressed by $y^2\; =\; ax$ (where $a$ is the parameter) we thence get $y\; =\; a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}$; and therefore $\backslash dot\{u\}\; =\; RmHB\; (=y\; \backslash times\; x)\; =\; a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}\; \backslash times\; x$: whence $u\; =\; \backslash frac\{1\}\{2\}\; \backslash times\; a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{3\}\{2\}\}\; =\; \backslash frac\{1\}\{2\}\; a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}\; \backslash times\; x\; =\; \backslash frac\{1\}\{2\}\; yx$ (because $a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}\; =\; y$) $=\; \backslash frac\{1\}\{2\}\; \backslash times\; AB\; \backslash times\; BR$: hence a parabola is $\backslash frac\{1\}\{2\}$ 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 CRNBC, whose corresponding abscissa AN is equal to AT $\backslash times$ 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 = $a$, AG = $s$, and AN = $u$, the fluxion of the area CDGB will be represented by $ys$, 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 fluxions.

Since, therefore, $ys$ is $=\; zu$, and $u\; =\; as$, by hypothesis, it follows, that $u\; =\; as$, and that the first equation (by substituting for $u$) will become $ys\; =\; a\; \backslash times\; z$, or $y\; =\; az$, or lastly $ys\; =\; zas$, that is, GD $\backslash times$ AG = NR $\backslash times$ 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 BG = $x$: then, since AG $(1+x)$

: AB (1) :: BC ($b$) : GD ($y$) we have $y\; =\; \backslash frac\{b\}\{1+x\}$, and consequently $\backslash dot\{u\}\; (=y\; \backslash times\; x)\; =\; \backslash frac\{bx\}\{1+x\}\; =\; b\; \backslash times\; x\; -\; xx\; +\; x^2x\; -\; x^3x\; +\; x^4x\; -\; \backslash dots$. Whence, BGDC, the area itself will be $=\; b\; \backslash times\; x\; -\; \backslash frac\{x^2\}\{2\}\; +\; \backslash frac\{x^3\}\{3\}\; -\; \backslash frac\{x^4\}\{4\}\; +\; \backslash frac\{x^5\}\{5\}\; -\; \backslash dots$, 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 $\backslash frac\{x\}\{\backslash sqrt\{x^2+a^2\}\}\; =\; \backslash text\{hyp.\; log.\; of\; \}\; x\; +\; \backslash sqrt\{x^2+a^2\}$;

of $\backslash frac\{x\}\{\backslash sqrt\{2ax+ax^2\}\}\; =\; \backslash text\{hyp.\; log.\; of\; \}\; ax\; +\; \backslash sqrt\{2ax+ax^2\}$;

of $\backslash frac\{2ax\}\{a^2-x^2\}\; =\; \backslash text\{hyp.\; log.\; of\; \}\; \backslash frac\{a+x\}\{a-x\}$;

and of $\backslash frac\{2ax\}\{x\backslash sqrt\{a^2-x^2\}\}\; =\; \backslash text\{hyp.\; log.\; of\; \}\; \backslash frac\{a-\backslash sqrt\{a^2-x^2\}\}\{a+\backslash sqrt\{a^2-x^2\}\}$.

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 = $\backslash dot\{z\}$; because the triangles OTC, OCS, are similar, OT : OC :: OC :: OS; whence OS = $\backslash frac\{a^2\}\{s\}$, and SA = $a\; -\; \backslash frac\{a^2\}\{s\}\; =\; a\; -\; \backslash sqrt\{\backslash frac\{a^2\}\{a^2+t^2\}\}$; whose

fluxion is $\backslash frac\{a^2\; \backslash dot\{t\}\}\{a^2+t^2\}$; and because the triangles OTC, dCe are similar, TC (=t) : TO ($=\backslash sqrt\{a^2+t^2\}$) :: Ce = $\backslash left(\backslash frac\{a^2\; \backslash dot\{t\}\}\{a^2+t^2\}\backslash right)$; Cd = $\backslash frac\{a^2\; \backslash dot\{t\}\}\{a^2+t^2\}\; =\; \backslash text\{fluxion\; of\; the\; curve\}$.

Now by converting this into an infinite series, we have the fluxion of the curve $=\; \backslash dot\{t\}\; -\; \backslash frac\{t^3\}\{a^2\}\; +\; \backslash frac\{t^5\}\{5a^4\}\; -\; \backslash frac\{t^7\}\{7a^6\}\; +\; \backslash frac\{t^9\}\{9a^8\}\; -\; \backslash dots$, and con-

sequently $z\; =\; t\; -\; \backslash frac\{t^3\}\{3a^2\}\; +\; \backslash frac\{t^5\}\{5a^4\}\; -\; \backslash frac\{t^7\}\{7a^6\}\; +\; \backslash frac\{t^9\}\{9a^8\}\; -\; \backslash dots\; =\; AR$.

Where, if (for example) take AR be supposed an arch of 30 degrees, and AO (to render the operation more easy) be put = unity, we shall have $t\; =\; \backslash sqrt\{1\; -\; \backslash frac\{1\}\{4\}\}\; =\; .5773502$ (because $Ob\backslash sqrt\{\backslash frac\{1\}\{4\}\}\; :\; AR\; (\backslash frac\{1\}\{2\})\; ::\; OA\; (1)\; :\; AT\; (t)\; =\; \backslash sqrt\{\backslash frac\{1\}\{4\}\}$) Whence,

etc.

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

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 in zoology. See NATURAL HISTORY.

FLY, in mechanics, a cross with leaden weights at its ends; or rather a heavy wheel at right angles to the axis of a windlass, jack, or the like; by means of which the force of the power, whatever it be, is not only preserved, but equally distributed in all parts of the revolution of the machine. See MECHANICS.

FLYING, the progressive motion of a bird, or other winged animal, in the air.

The parts of birds chiefly concerned in flying are

EXAMP. Let it be proposed to find the content of a cone ABC, fig. 10.

Put the given altitude (AD) of the cone = $a$, and the semi-diameter (BD) of its base = $b$, the solid = $s$, its fluxion = $\backslash dot\{s\}$, and the area of a circle, whose radius is unity, = $p$: then the distance (AF) of the circle EG, from the vertex A, being denoted by $x$, &c. we have, by similar triangles, as $a\; :\; b\; ::\; x\; :\; EF\; (y)\; =\; \backslash frac\{bx\}\{a\}$. Whence in this case, $\backslash dot\{s\}$

which, when $x=a$ ($=AD$) gives $\backslash frac\{p\; b^2\; a\}\{3\}$ ($=p\; \backslash times\; BD^2\; \backslash times\; \backslash frac\{1\}\{3\}\; AD$) for the content of the whole cone ABC: which appears from hence to be just $\backslash frac\{1\}\{3\}$ of a cylinder of the same base and altitude.

PROB. 4. To compute the surface of any solid body.

The fluxion of the surface of the solid is equal to the periphery of the surface, by whose motion the solid is generated, multiplied by its velocity on the edge of the solid, and the computation is made as in the foregoing.

EXAMP. Fig. 11. Let it be proposed to determine the convex superficies of a cone ABC.

Then, the semi-diameter of the base (BD, or CD) being put = $b$, the slanting line, or hypotenuse AC = $c$, and FH (parallel to DC) = $y$, AG = $z$, the surface = $w$, its fluxion = $\backslash dot\{w\}$, and $p$ = the periphery of a circle whose diameter is unity, we shall, from the similarity of the triangles ADC and Hmb, have

$\backslash frac\{2pcy^2\}\{b\}$; and consequently $w\; =\; \backslash frac\{p\; c\; y^3\}\{b\}$. This, when $y=b$, becomes $=\; p\; c\; b\; =\; p\; \backslash times\; DC\; \backslash times\; AC\; =$ the convex superficies of the whole cone ABC: which therefore is equal to a rectangle under half the circumference of the base and the slanting line.

The method of fluxions is also applied to find the centres of gravities, and oscillation of different bodies; to determine the paths described by projectiles and bodies acted on by central forces, with the laws of centripetal force in different curves; the retardations given to motions performed in resisting media; the attractions of bodies under different forms; the direction of wind, which has the greatest effect on an engine; and to solve many other curious and useful problems.

the wings, by which they are sustained or wafted along. The tail, Messrs Willughby, Ray, and many others, imagine to be principally employed in steering and turning the body in the air, as a rudder: but Borelli has put it beyond all doubt, that this is the least use of it, which is chiefly to assist the bird in its ascent and descent in the air; and to obviate the vacillations of the body and wings: for, as to turning to this or that sides, it is performed by the wings and inclinations of the body, and but very little by the help of the tail. The flying