404 Mr DAVIES on the Equations of Loci 



. Again, to find a, we have from (3) the equations 



or cos' 1 - cos- 1 =6 a Q. ........................... (5.) 



r. r u 



Take cosine-function of these, square, and transpose : which gives, after 

 slight reduction, 



v rj 9,r l r ll cosQ ll 6, + r* 

 Insert these values of /3 from (4) in equation (2), and we have 



= cos (B cos 6 + sin /3 sin Q 

 r 



= -+- cos 6 (r, sin 6, r n sin 6,) ZjZ sin 6 (r, cos 0, r,, cos $) 

 ^ r, z 2 r, r t cos 0,, Q, + r,, 2 



and from equation (6) inserting the value of a, we find finally 



(7.) 



r, r a sin > 6, ( +- cos 6 (?-, sin 0, r y/ sin $ . 



r ( :+: sin (r, cos 6, r a cos a 



or by division, 



, sn z y r sn 



If now we keep in mind that , , and are the same things that cot <p, 



' ' / r n 



cot a, and cot a,, on the sphere become when the sphere is infinite, we get 

 precisely the same result as in my paper, (IV. 10). That is 



~ 



- sin tf-t-f. sin 



^=^ -^ do-) 



is the form which may be considered as the extreme case of that result, the 

 sphere becoming then infinite. 



