OF A CURVE AND ITS APPLICATION. 153 



45. I will take some other cases of the intrinsic equation. 



ds 

 Let — = a sin + 6 sin 20 = a sin <p + 26 sin cos 0. 



It is evident that there is a cusp when sin = 0, and when cos = . 



Hence the form of the curve is that in figure 7. 



We have dy = ds sin = a sin 2 0d0 + 2ft sin 2 cos 0. 



But /sin 2 d0 = ^0 - \ sin cos 0. 



a a . 26 . 



Therefore y = - — sin cos + — sin 3 0. 



From = to = 7r, this = — = AM. 



Also * = Jds cos = /(a sin cos + 6 sin 20 cos 0) dd>. 

 = /(a sin cos + 26 sin cos 2 0) d0. 



= C COS^ COS d G> 



2 r 3 r 



a , 26 



= - (1 — cos 2 0) + — (1 — cos 3 0). 



46 

 When = tt, this = — = MC. 



Hence AM depends on a alone, and MC on 6 alone. 



ds 



46. Again, let — — - = a sin + 6 sin 3 = a sin + 6 (3 sin - 4 sin 3 0) 



= {a + 36) sin 0—46 sin 3 0. 



There is a cusp when sin = 0, and when sin 2 = — , 



'46 



dy = ds sin = (a + 36) sin 2 - 46 sin 4 0, 



dx = ds cos cp = (a + 36) sin cos0 - 46 sin 3 cos 0, 



(a + 3 6) . , 

 X = sin 2 — 6 sin 4 0. 



When = - , # = : when = ir, * = 0. Form as in figure 8. 



47. All such curves as have equations of these forms may be called generally cycloidal 

 curves. They have cusps and sinuses which, after going through a certain cycle, recur. 



Vol. IX. Part I. 20 



