traced upon the Surface of the Sphere. 325 



applied to the question before us, it requires, amongst other conditions, that 

 the denominators in the two terms which compose the value of d 6 in equa- 

 tion (4) shall become identical*. This is quite independent of the curve 

 being algebraic or transcendental (that is, as before stated, independent of 



7T 



the ratio of sin R to sin r\ by making p = -, which reduces the expres- 



sc 



sion to 



- sin r s\n(f)d<p cos R HZ r cosec <pd(f) 



"sinR Vsin*R:r cos*0 Vsin*RTr cos*< 



sin R cos Rdzr sin r sin* (f> 

 sin R sin (f> 



V sin* R+r sin 2 (j) 



Inserting this in ^/sin* <p d 6* + d <p 2 , we obtain the element of epi- 

 cycloidal arc 



smd)d(b /sin* r sin* (ft + sin 8 R 2 sin R sin r cos Rr 

 L = sinR * sin</> cos'Rztr 



The radical of (6) can only become rational when 



2 sin R sin r cos R + r sin*R = sin*r cos 2 Rr (7.) 



This divides itself into two cases, which must be considered separately, 

 namely, according as R db r or R r is taken. 



1. TakeR + r: then expanding and reducing with respect to sinR, 

 we have 



* Perhaps it would be more accurate to say the one should be a multiple of the other 

 by a constant factor : but it will be found that this introduces a complexity into the ex- 

 pression, which is fatal to the subsequent rationalization of the general formula for the arc. 

 The result of such a supposition is an imaginary quadratic. However, as upon this hinges 

 the only doubt respecting the general conclusion in the subsequent investigation, I shall 

 enter upon it with all requisite detail on some future occasion. 



VOL. XII. PART 1. T t 



