n m log a n'n nx arc to tan." 1 + m log. a 1⋅n- 2 x 3 m3 log. a n n.n 1 r m2 · log.1 a + &c. &c.) m + 1 34. y log. y + sy ON THE CORRECTION OF FLUENTS 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 ris n r 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 m+12 2-3 I + &c. n.log. a ys its fluent is r or r 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 a+ x 4 + where which should also The true fluent of this quantity, however, and many others may be found without correction; in the present case (ya+r3 i) if a + x be expanded, we have y a 3 a2 x + 3 a x2 i + x3 i, whose fluent is y = a3 x + a + x as 4 3 a2 x2 2 4 -- 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. Let y2 be the proposed fluxion, then its Juent is y = 23 the vessel will hold, or pry= = 4 c 3 3 Here when y = 0, 3 is a maximum and consequently — Again, let j = x^i, then y which, corrected, becomes y = n+1; n + 1. pr 8 c, and pr2 y = 4 c, x = 2y; whence y is also known, and it appears hence too that the diameter of the base must be just double the altitude. 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 N E 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 Gistance CN. Then we have b::: c : Example 3.-From a given point P, in the Hence M N2 CM2 + CN2 — 2 CM ·CE= transverse axis of an ellipse, to draw PB, the shortest line to the curve: B the fluxion of which put=0, gives + 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 Ea e, and TED are similar, and therefore e a a E:: ED: DT. Orỳ: ¿::y: y=DT, the subtangent; a being the absciss ÁD, and y the ordinate DE. r2s 5√52 Any of these quantities may be expanded in a series, and the fluent of each term being taken, a general value of z will be obtained. We shall gs t take as an example i = This form of the series, however, is one of very slow convergency, so that a great many terms of it must be collected before a result of sufficient practical accuracy can be obtained. But it may easily be transformed into series of almost any required degree of convergency. The following are amongst the most useful forms that have yet been discovered; A representing the Example 1.-To draw a tangent to a parabole, circumference to radius unity, and a, ß, y, &c., whose equation is a x = y2. the preceding terms in each series :— 2 ax y i 2 y y a 2 = a Here a 2 yj; whence = =2; or the subtangent is double the corresvonding absciss. Example 2.-Draw a tangent to the cissoid of 1st A = Diocles, whose equation is y 3 = 10 a-x. 3 ax2 i 2 r3 r 8 Y and conse + + &c. 9.10 2x 3 a- ལྤ 3ax2-23 16 Y + 9.10 2d 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 A E; 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 s = √√ ï1+j3. Therefore substituting for y its value in terms of 1, and taking the fluent, the value of z is obtained. n3 y n'y 2n-2 4 n 2 n - 1.2 a 4 n 3.8 a 6 n 5 &c. 6 6 n no y 6 n 5.16 a 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. ai √2+ja2= i=1, and 2 cyiaci, 2 whose fluent is a cx, the value of the spherical surface. But a c is the circumference of the generating circle; hence the surface of any segment is equal to the circumference of a great the segment; and, when this versed sine is the circle, multiplied by the versed sine; or height of whole diameter, the expression is c a2, or four times the area of a great circle of the sphere. Example 2.--Let the proposed curve surface 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 rated by the plane itself. Hence, if x, y, and c, the surface bounds may be conceived to be generepresent 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. Here ab::: y = c b2 x2 x whence cy'i= b x a ; 3 a2 which when whose fluent is = a, becomes cb2 a 3 or one-third of a cylin der, having the same base and altitude. Example 2.-Let the proposed body be a spheroid, the tranverse and conjugate of whose generating ellipse are a and b. b2 a 2 By the nature of the curve (see CONICS) y2 2, whence cy2 x = 45. A surface may be conceived to be generated by the circumference of a plane moving forward, and expanding at the same instant; fore the fluxion of the surface is equal to the whose fluent is fluxion of the curve, in which the expanding circumference moves forward at any instant, multi-a isplied 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, the abscissor, ythe ordinate, and * the curve in which the expanding circumference moves forward; then 2 c y = the circumference, and 2 c y ¿ — 2 c y √ x2 + j2 = 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 c b2. a2 23 3 = (a); which when 2 the content of the whole sphe roid. And if a = b, the spheroid becomes a cas 6 its circumscribing cylinder, for the solidity of the is ca b2 4 TO FIND THE POINTS OF CONTRARY FLEXure of 47. It is evident when a curve is concave as of DE; hence the equation may be put in this form GC-ii GEy-yy; or GC contrary fluxion or is a constant quantity, and consequently its fluxion is 0. Therefore, if GEy ï3 + j3 3; and if each of the terms of this equation be respectively multiplied by the equivalent expressions GC GE' from the given equation the value of or be found, the fluxion of that value will give an equation, from which the relation of x and y at the point of contrary flexure may be found. Example 1.-Required the point of inflexion in a curve, whose equation is a r2 = a2y+ry. This equation in fluxions is 2 a x x = a2 j + a2 + x2 2 x y i + x2 j, whence —— the y 2αx- 2xy fluxion of which made =0, gives 2 r i (ar-ry) = (a2 + x3) · (a i j — ż y), and this again a2 + x3 gives ; which being equa ; whence ra√, a— y a r3 +x and consequently y =π2++2 = ta. 3 ; a general expression for r in any curve. But as neither a ory may be considered as varying uniformly, or either r or y may be considered as 0, we may have r according as y or is con 23 or sidered constant. y x 23 -xy Example.-Required the radius of curvature at the joints of an ellipse, at the point corresponding to the absciss and ordinater and y, the equation of the curve being a2 y2 — c2 a x — By taking the first and second fluxions of the given equation we have 2 a2 y j= c2 ¿ · a — 2 x, and 2 a3 j2 + 2 a2 y ÿ = 2 c2 2, considering ca i · a 2 x, and 2 ay as constant; whence y= ; which, by substituting the values of y and y, become y= } a2 i + } x2 i √ a 2 y= a3 j 2 + c2 ¿ 3 a2 y The cra 2 x 2a/ax Auction of this put= 0, and reduced, gives r= a (√3—1) 4√12 TO FIND THE RADIUS OF CURVATURE OF 48. The radius of curvature is that of a circle having the same curvature as that of the curve at any proposed point; the general method of finding the radius of this equicurve circle may be thus explained. D Let AD and DE be the absciss and ordinate to the curve A E, EC the radius of the equicurve circle at E, consequently perpendicular to the curve at E. From C as a centre, with radius CE, describe the circular arc B Ex; draw C B parallel to AD, and let ED produced, meet BC in G; draw Ed, and ed parallel to ED and AD, to represent the fluxions of AD and DE. Put AD=r, ED=y, and A E = 2. Then by similar triangles GC: GE:: j: i, or GC GEj. Whence by fluxing GCx + GE +GE j. But GC BG, therefore, GC r - ·BG · GEy +GE y. Now is the fluxion of BG, as well as of AD, and y is the fluxion G E as well a√ a x-x3 (√ ż 2 + j 2) = 4 ax hence 2 4r, and r c3 a2 + a2 2 a 2 a c which when a and c are equal becomes as it a ought simply, the ellipse in that case degenerating into a circle. TO FIND THE INVOLUTES AND EVOLUTES OF CURVES. 49. If a thread wrapped close round a curve were fastened at one end, and unwound from the other in the plane of the curve, the thread being always kept stretched, the end of the thread in winding off will describe a curve which is called the involute, that from which the thread is unwound being the evolute. Now it is obvious that the length of the thread wound off will be the radius of curvature of the involute at the instant, and also that it will at that point be perpendicular to the involute; and that the evolute will be the locus of the centres of the radii of curvature at every point of the curve. |