on tivo Spherical Surfaces. 59 



We in fact, from the foregoing relations, at once obtain 



a 2 b 2 L 



b 2 



rk 7 / & X 



^ c — x~~ l \a 1 x + fii u 2 x + P 2 '"} 



v c — x icix + di c 2 x + d 2 j 



To satisfy the first equation we must have M = aL ; viz., this 

 being so, the equation becomes 



, b 2 , / b 2 \ ahh 

 ac{>x+ <£>( )= — j', 



c — x \c — xj c o x + d 



or, since c x-\-d = l, the equation will be satisfied if only 

 «L = 1, whence also M = l. And the second equation will be 



27.2T 



satisfied if only =bM ; viz., substituting for L, M their 



value, we find w = ab. 



Supposing, in like manner, that h = 0, g retaining its 

 proper value, Ave find a like solution for the two equations ; 

 and by simply adding the solutions thus obtained, we have a 

 solution of the original two equations 



a4>(*)+ —&(—)=h, 



r / c—x \c—x) 



c 

 viz. the solution is 



a ^j(^y^)=9; 



^ ' a \ c ^ + d Ci^ + di "j t* X&icc + hi a 2 «£ + b 2 'i 



*(*)- h f ah i w i i+gj l + ah + \ 



* W a^+^ + ^4.^ + ■■•]■ + 5 l 7o « + 8o 7i^ + Si i 



We have a general solution containing an arbitrary constant 

 P by adding to the foregoing values 



for (fix a term ~Pb(a — b) 



\/fl 2 (c — x) — x(c 2 — b' 

 and for <§x a term 



Fa(b-a) 



V 6 2 (c — x) — x(c 2 — a 2 — cxy 



as may be easily verified if we observe that the function 

 a 2 (c — x) — x(c 2 — b 2 — ex), 



