traced upon the Surface of the Sphere. 327 



If w consider r the arbitrary, we shall have the equations 



{sin 8 r (5 sin 4 R + 2 sin 2 R 1) + sin r (sin 2 K + 4 sin 4 K) 

 + sin 4 r (7 sin 4 R 2 sin 2 R + 1) + sin* r (8 sin 4 R 2 sin 1 R) 

 + sin 4 R = (12.) 



for Ur, sin 4 r 2 sin* R sin 2 r + sin 4 R = ....(13.) 



Now, that the proposed object may be attainable at all, we must have 

 one of the following sets of conditions fulfilled. 



From equation (10.) 



0= sin 4 rcos 4 r a, 



0= sin r cos r (1 + sin 2 r) b, I (14.) 



= (1 + sin 2 r) 2 {(1 + sin 2 r) 2 + 4 sin*r cos 2 r] c, 



From equation (11.) 



n. i 



(15.) 



0= cos 4 rsin 2 r b a 



From equation (13.) 



0=5 sin 4 R + 2sin 2 R 1 a,, 



0= 4sin 4 R + sin 2 R b, H 



0=7sin 4 R 2sin*R + l c,A (16.) 



0=8sin 4 R 2sin 2 R d ltl 



0=sin 4 R e,, 



From equation (14.) 



0== 1 o r 



0= 2sin 2 R 6 n 



0= sin 4 R c iv 



In the set (14), equation a, is separately fulfilled (and fulfilled alone) by 

 sin r = 0, and cos r = 0. 



Equation b, is fulfilled also by each of these : but equation c, is not ful- 

 filled by either of them. Hence, so long as in the spherical epicycloid we 

 employ R + r, there is no value of r that will leave R arbitrary, and yet 

 fulfil the conditions required to render (6) rational. 



T t 2 



