22 PHILOSOPHICAL TRANSACTIONS. [aNNO l/Oa. 



radius be r, the arch c, and the ordinate of the generating circle to be y, whose 

 diameter may be represented by ak, and the centre posited between l and k. 

 Then calling fd the ordinate of the cycloid, o, and the rest as before, the 

 equation of the curve is aa-=. yy -f 2cy -|-cc, and therefore zz = {aa -f- nn — 

 *lnx -f- XX =^)yy + 2cy -|- cc -f- «n — liix -|- xx, and 2 being determined for 



an extreme, 23/3^ -f Icy -f '^if'c -j- 2cc — 2nx -j- 2xi' = 0. But y = , and 



c = — ; then substituting these values, and duly reducing the equation, we shall 

 have 2r — oc A V '^r A = 2n — - 2 a?, and therefore 



1r — X A ~ — = n — jf = DL the subnormal. 



The incomparable Dr. Barrow makes use of the subtangent as already known, 

 to determine the maximum and minimum. And Mr. Newentiit, in his Ana- 

 lysis of Infinites, has done the same after him. But since the maxima and 

 minima may be found by many other methods, in which nothing need be pre- 

 supposed about the tangents of curves, it is plain that we may safely proceed 

 from the maxima and minima to investigate the method of tangents. 



Corol. 1. Ingoing over again the foregoing examples, it will appear from 

 each, that lyif — Inx + Ixx = 0, by putting instead of n in this equation its 

 value derived from the nature of the curve. For example, in the hyperboloids 



Six ^ — Ixx + —i'x ^ 4" ^jpi* = O, which appears by inspection. And 



the same will appear to be true in other examples, without any demon- 

 stration. 



Corol. 2. From the invention of the subnormals we may easily determine the 

 greatest and least ordinates of curves. In which matter I shall add, if the sub- 

 normal belonging to any point of the curve be put equal to nothing, we shall 

 have the ordinate of that curve determined to be an extreme. And it will be 

 the greatest, if it is on the concave side of the curve, but the least if it is on 

 the convex side. For example, in the circle, making the subnormal =: /, it will 

 be / = r — X. Let r — * = O, then r = *, and y = r ; that is, the greatest 



ordinate is equal to the radius. In like manner, in the ellipsis / = --}-—; let 



L — r= O, then rq = 2rXt or * = \q. Therefore yy = -j-ry, equal to a 



2 q 



fourth part of the figure, as they call it, or the square of the conjugate semi- 

 axis, and therefore the greatest y is equal to that semiaxis. And the same 

 method may be used in other curves. Let the subnormal be found from the 

 given equation, and making that equal to nothing, we shall have the ordinate 



