152 Dr. whewells second memoir on the intrinsic equation 



ds 

 And before — - = a W (l+m 2 + 2m cos 8), 

 ad 



. ds a (1 + m 2 + 2 m cos 9)% 



whence — = ; . . . (3) 



d(p 1 + m cost) 



which is the intrinsic (differential) equation to the protracted cycloid. 



To find s in terms of <p, we have (1) and (3). The integrations would give an elliptic 

 transcendant. But we may integrate by the following construction. 



43. Let ab, fig. 6 be an elliptical quadrant, in which 



ac = a (1 + m) = DA, be = a (l - tn) 



and let ad be a quadrant of a circle with radius ac, and let dco = ^9, mpo an ordinate to the 

 ellipse and circle. 



Then 



Hence 



(1 + m) 2 

 But the differential of the arc bp is to the differential of the arc do as tp to do. 



And it is evident that the differential of do is ac — , or d9. Hence 



2 2 



d .bp = - d9<\/ (1 +m 2 + 2m cos 9). 



But by (2), d.AP = add ^/(l +m 2 + 2m cos 9). 



Hence AP, fig. 5, = 2&p, fig. 6, since they begin together. 



44. Now when m becomes 1, the protracted cycloid becomes the ordinary cycloid, and the 

 loop vanishes. But in this case cb, = a (l — m), also vanishes, and the arc which was 2bp 

 becomes 2cm, and the increment becomes negative when we go beyond a. Therefore in this 

 case we have a confirmation of the mode in which we have measured s. 



The same would be the case if we had begun with the contracted cycloid or trochoid. The 

 equations are the same as in the case just examined except that m is > 1. The curve becomes 

 the common cycloid when m = 1, and the elliptical integration changes its nature for this case. 



