Of GREATEST ATTRACTION. 191 



From C (Fig. 1.) draw CE perpendicular to AB ; let ABzza, 

 AE=x, ECzzy. We have found AB 2 XAE:=AC 3 , that is, a x— 



{x % +y) T , or a 4 x 2 zz (x 2 -{-y 2 ) , which is an equation to a line 

 of the 6th order. 



4 2 42 



To have y in terms of x, x -{-y 1 zza 3 x 3 , y z zz a 1 x T — .ry 



and y zz x 3 V a* — x 3 . 



Hence y zz o, both when x zz o, and when xzz a. Alfo if S? 

 be fuppofed greater than #, j/ is impoffible. No part of the 

 curve, therefore, lies beyond B. 



The parts of the curve on oppofite fides of the line AB, are 

 fimilar and equal, becaufe the pofitive and negative values of y 

 are equal. There is alfo another part of the curve on the fide 

 of A, oppofite to B, fimilar and equal to ACB ; for the values 

 of y are the. fame whether x be pofitive or negative. 



III. 



The curve may eafily be conftructed without having recourfe 

 to the value of y juft obtained. 



Let AB=#, (Fig. 1.) AC = z, and the angle BAC = (p. 



Then AE = AC X cof<p zz z cof <p, and fo a 'z cofp zz z l , or 



a 2 cof <p zzz* ; hence zzza\f cof <p. 



From this formula a value of AC or % may be found, if (p or 

 the angle BAC be given ', and if it be required to find z in 

 numbers, it may be conveniently calculated from this expref- 

 fion. A geometrical conftru&ion may alfo be eafily derived 

 from it. For if with the radius AB, a circle BFH be defcribed 

 from the centre A j if AC be produced to meet the circumfe- 

 rence 



