Dr WHEWELL'S second memoir on the intrinsic equation, etc. 151 



points Y, Y' which correspond to the vertex D of the evolute, will coincide ; and the form of 



a 2 

 the curve will be as in figure 2. This is if — = — . 



7T 



If - be greater than this, the larger sinuses of the curve intersect, as in figure 3. 



41. The cusps Z, Z' approach nearer to each other, as - approaches nearer and nearer 



to 1. When a = b, the two cusps disappear, and the curve becomes looped. Yet at the 

 vertex of the loop the radius of curvature is = 0. 



When a > b, the curve is looped, and the radius of curvature never vanishes. 



In this example the loop arises from the evanescence of two cusps ; and if we take the 

 order of changes inversely, the evanescence of the loop gives rise to two cusps. In the curve 

 s = ad) + 6 sin 0, the smaller sinus between two cusps is negative, and when a vanishes, the 

 smaller and larger sinus become equal, which is the case of the cycloid ; and hence in that 

 case the alternate sinuses are negative. 



42. The cycloid may however be considered as the ultimate form of a looped curve, 

 namely, of the protracted cycloid. If while the circle VB, fig. 5, rolls upon the straight line 

 DB, a point P in the radius QC produced describe a curve AP, this curve is the protracted 

 cycloid. 



To find the intrinsic equation to the protracted cycloid, let DA be the original position of 

 QP; PM, and CE, perpendicular on AD; AM = a, MP = y, VCP = 9. And let CP = a, 

 CB = ma. 



Then x = AM = EA - EM = CP - CN = a - a cos 9, 



y = MP = MN + NP = DB + NP=BQ+NP = ma9 + a sin 9. 



Also the curve at P is perpendicular to BP : and the curve at A is perpendicular to DA. 

 Hence the angle between BP and DA, that is, angle PBN = (p. 



wu NP s ™ e 



W hence tan d> = = (1) 



r BN m + cos 9 w 



We have dm = a sin 9 . d9, dy = a (cos 9 + m) d9 : whence 



ds = y/(da? + dtf) m ad9^/(\ + wi 8 + 2m cos 9). ... (2) 



Also, from (l), sec 2 <b . dd> = ( ^ + mcos ^ dd 

 T T (m + cos 9y 



i » , o , 1 + m 2 + 2mcos9 . , ^ 



but sec' m=l+ tan - * d> = — — : by (1) 



T T (m + cos 9y 



d9 l + m s -+2mcos# 



whence — - = -p. . 



a0 1 + mcost/ 



