on two Spherical Surfaces. 57 



Observe that these equations give, as they ought to do, 



a x + h =x, c # + d = l, a 1 ^? + b 1 =a^ + b, c 1 ^? + d 1 = c^? + d; 



and similarly 



a x + /3 = x, y x + 8 =l, « 1 ^ + /9 1 = a^ + /3, 7 1 ^ + S 1 = 7^-f 5. 



a 2 



Substituting in the first two equations in place of x, 



p c—x r 



and in the second two equations in place of x, we obtain 



the following results which will be useful : — 

 a n a 2 + b„ (c — x) = a 2 (y n x + Sn), 

 c n a 2 + d n (c-x)=p(a n+1 x + /3 n+1 ), 

 a n h 2 + &n(c —x) — b 2 (c n x + d n ), 

 y b 2 + S n (c—x) = — 2 (a n+l x + b n+1 ), 



the last two of which are obtained from the first two by a mere 

 interchange of letters ; it will therefore be sufficient to prove 

 the first and second equations. 

 For the first equation we have 



a w a 2 + b w (c-^)=^^(^) W " 1 {(^ +1 --l)[a« 2 + b(c-^)] 



+ (\ n -X)[-da 2 + b(c-x)']}, 

 where the term in { } is = 



(\ n+l -l)l-a^a 2 c(c-x)^+(\ n -\)[a\b 2 -c 2 ) + a 2 c{c-x)-]; 

 viz. this is 



= a 2 {(\ n + i -l)(c 2 -a 2 -cx) + (X n -X)(b 2 -cx)}; 

 or it is 



= a 2 {(\ n + 1 -l)(yx + 8) +(\n-X)(yx-*)}, 



whence the relation in question. 



The proof of the second equation is a little more compli- 

 cated : we have 



c n a 2 + d n (c-x) = ^- i (0^y~ l {(^ +1 -1) [ca 2 + d(c-*)] 



+ (X w -X)[ca 2 -a(c-^)][ 

 where the term in j j- is = 



(X-+ 1 — l)i-y+(a»-^(«-«)] + ^"-X)[-«^ + ^(c-«]. 



