660 Dr. WHEWELL, ON THE INTRINSIC EQUATION OF 



5. Hence s, s", &c. indicating the successive evolutes, p, p, p", &c. the successive radii 

 of curvature, we have, 



I dS f, II ^^' f^l <'*"' fylll 



d<p dip dtp 



ds , ds „ ds' 

 P^d^' P^d^' P =d^'^"- 



6. Also we may in like manner find the successive involutes s„ Sj) «3) &c. For we have 



-^ = s'+C=f(<p) + C,s =/ ((p) + C<p. 



So ^ = «+C, =/(^)+ C^ + C, 



s,=f,{<p) + C% + C,<p. 



Hence s is known in function of (p, and therefore the curve known. And in like manner s„, S3, &c., 

 if these be the arcs of the successive involutes. 



In Fig. 4, CJR, BQ, AP are successive involutes of DS. 



7. It is evident that the intrinsic equation to the circle is 



8 = ad), a being the radius. 



Also for the equiangular spiral, since the curve from its origin is everywhere similar to itself, 

 the radius of curvature is proportional to the whole arc. Hence 



ds 



—7 = /n« ; whence s = o""*', if s be measured from the pole. 



dm 



If 8 and (p vanish at the same time, s = a (e""* - I). 



We shall afterwards give general formula^ for obtaining the intrinsic equation from the ordinary 

 coordinate equation, and reversely. But the operation of our method will be better seen by first 

 taking some special cases. 



Of Cycloids, Ejiici/cloids, and Hypocychids. 



8. In the Cycloid, if VB, Fig. 5, be the diameter of the generating circle, rolling on the 

 straight line DB from the initial position AD, when it is perpendicular to DB, and P the describ- 

 ing point at that time, PQ being the diameter, by the mode of description, the arc BQ = ED. 

 But the curve at P is perpendicular to PB ; and if be the angle of deflexion, <p = VBP, and 

 20= VCP. Hence chord VP = 2b sin <p, if h be the radius of the circle. And the arc AP 

 = 2 chord VP. Hence the intrinsic equation to the cycloid is 



« = 4 6 sin d». 



When (p becomes a right angle, « becomes a maximum. At this point there is a cusp (Z), and the 

 added part of s after this is negative ; and so continues, till = 3 right angles, where there is 

 another cusp (Z), and the added part of s becomes positive ; and so on. 



