ax 6 a 40 a3 x7 112 a 5 x 1152 a7 &c. 38. There have been collected by several authors a great many forms of fluxions, with the Again, let it be required to find the fluent of corresponding fluents. They may often save much labor in finding the fluents of complicated expressions, when a fluent is to be found from a fluxionwhich either agrees with, or has an assignable relation to, a fluxion on those collections. They serve in this case much the same purpose to the analyst, that logarithmic tables do to the compeWe give the following as the forms which are of most frequent occurrence in practice, and refer for the most extensive collection with which we are acquainted, to a work on the subject by Meyer Hirsch, which has been lately translated into English: ter. n ON THE CORRECTION OF FLUENTS n—i m+1 2 √ a m log. a n- 2 r 3 m3 log. 3 a n'n-1 m + N13 n log. r log. &c.) x + &c. n.log. a yx N-2 n n 3 m + 1 log. 40. Though, by the rules which have been given for finding fluxions, the fluxion of any fluent may be found, and by a reverse operation the fluent may in most cases be found from the fluxion; yet the fluent so found may often require to be increased or diminished by some constant quantity depending on the nature of the problem under consideration. For example, the fluxion of r is n x i, and the fluxion of ra is the same quantity; we cannot therefore affirm without reference to the nature of the problem in which the fluxion n x i arises, whether its fluent is r or r +a. The most direct and simple method of finding whether a fluent does or does not require correction, and the amount of that correction, if any, when y = 0, is to see what the variable part of the fluent (the Example 2.—r and y commencing together, let the true fluent of ja be required. where By the common rules y = a+ x a + x which should also 0, hence y= rected fluent. The true fluent of this quantity, however, and many others may be found without correction; in the present case (j = a + x3 i) if a + x 3 be expanded, we have ya3 + 3 a2 x x + 3 a x2 x + x3 i, whose fluent is ya x + 3 a2 x2 2 3 4 a + x 4 3 as before. +ax + In the preceding examples x and y are supposed to be equal to nothing at the same time; but in the solution of problems this will often not be the case. Thus, though the sine and the tangent of an arc are nothing, when the arc itself is nothing, yet the secant and the cosine are then equal to the radius. We shall therefore add an example or two, in which when y = 0, x is equal to a given quantity a. Letja be the proposed fluxion, then its fluent is y = Here when y = 0, p r — x2) + x2. Hence, by putting the fluxion p-q. =0, and reducing, we get x= 2 62 a2 x 3 = 3 3 4 c I + is a maximum and consequently — Pri 2 = 0, or x=23 P + ; and as pr 8 c, and pay 4 c, x = 2y; whence is also known, and it appears hence too that the diameter of the base must be just double the altitude. y Example 5.-If two bodies move at the same time from two given points A and B, and proceed uniformly with given velocities in given directions AP and BQ; required their positions, when they are nearest to each other. Let M and N be any two contemporary positions of the bodies, and upon AP let fall the perpendiculars NE and BD: produce QB to meet AP in C, and draw MN. Let the velocity in BQ be to that in AP, as n to m, and let AC, BC, and CD (which are also given) be denoted by a, b, and c, respectively, and put the variable Cistance CN r. Then we have b¦ ¦ ¢ ¦ Q Let the required fraction, the rˆ — r is 1 CE= b mx-mb to be a maximum, or nx i = 0, n E D and n: m :: BN (r—b); AM= m b ; and hence CM = a + Hence MN2 CM2 + CN2 — 2 CM · CE= 2 d mi m 2 cd r 2cm2 d2. + + n n2 b the fluxion of which put=0, gives n b, 2 d mi n If A E be a curve, let it be required to draw a tangent TE at any point E. Draw the ordinate DE, and another da e indefinitely near to it, meeting the curve, or the tangent produced in e, and draw E a parallel to the axis AD. Then the triangles E a e, and TED are similar, and therefore e a a E:: ED: DT. Orý: :: y: DT, the subtangent; r being the absciss AD, and y the ordinate DE. Example 1.-To draw a tangent to a parabole, whose equation is a x = y2. y i 2y2_2 ax Here a i2yj; whence = y a = a 2r; or the subtangent is double the corresDonding absciss. Example 2.-Draw a tangent to the cissoid of 1st A Diocles, whose equation is y2 = a J. 10 2 1+ 3.10 3 ar i 2 r3 i Y Here 2 y j = + &c. 9.10 a TO DETERMINE THE LENGTHS OF CURVES 43. In the annexed figure E a, e a, and E e, are simultaneous increments of x, y, and z, or of the absciss AD, the ordinate D E, and the curve AE; and the triangle Eae is (see article 42) similar to TED; it may therefore be considered as a right-angled triangle. Hence, = i2 + j2, or i = √√ i 2 + j 3· Therefore substituting for y its value in terms of a, and taking the fluent, the value of z is obtained. 9. 100, 12 B + 7.10 2 n 2721 ny 4π-3 2 n 2 1.2 a n 3 4 n 8 a 6 n- 5 6 n 6 n- 5.16 a &c. + But when 2 n − 2 is either unity, or an aliquot part of it, this series will always terminate, and consequently the length of the arc will be accurately obtained from it. a √ *2+j2 = i = 21, and 2 c y i = sci, TO FIND THE AREAS OF CURVES, WHOSE EQUA- be that of a parabolic conoid. TO FIND THE SOLID CONTENTS OF BODIES. 46. As a curve surface may be conceived to be generated by the expanding circumference of a plane moving forward, as the solid itself which the surface bounds may be conceived to be generated by the plane itself. Hence, if x, y, and c, represent the same things as they did in art. 44, we have c y2 i for the fluxion of the solid, and the fluent of this quantity will be the required solid. Example 1.-Required the solid content of a cone, whose altitude is a, and base b. b r a xx.1 a a2 c b2 a 3 whose fluent gives r√ a r· 3 a2 or one-third of a cylin for the area of the der, having the same base and altitude. TO FIND THE SURFACES OF SOLIDS. 45. A surface may be conceived to be generated by the circumference of a plane moving forward, and expanding at the same instant; therefore the fluxion of the surface is equal to the fluxion of the curve, in which the expanding circumference moves forward at any instant, multiplied by the periphery of the variable circumference at the same instant; and the fluent of this fluxion is the value of the generated surface. If c the circumference of a circle, whose diameter is 1, a the abscissor, ythe ordinate, and ≈ the curve in which the expanding circumference moves forward; then 2 cy=the circumference, and 2 c y ¿ = 2 c y √ 2+2 the fluxion of the surface S, and consequently by taking the fluent, S is obtained. Example 1.-Let the proposed curve surface be a sphere. In this case y√ a x − x 3, Example 2.-Let the proposed body be a spheroid, the tranverse and conjugate of whose generating ellipse are a and b. 62 By the nature of the curve (see CONICS) = • a &— x2, whence c yi a c b2. ·ari-fi, a2 ca b2 ra isthe content of the whole sphe 6 roid. And if ab, the spheroid becomes a c a3 sphere, whose solidity is 6 Hence a sphere, or a spheroid, is two-thirds of TO FIND THE POINTS OF CONTRARY FLEXURE OF 47. It is evident when a curve is concave towards its area, that the fluxion of the ordinate decreases with respect to the fluxion of the absciss; and the contrary when the curve is conof vex towards its axis; hence, at the point |