[706 
706] 
ON TWO SPHERICAL SURFACES. 
5 
a (c-«)]}, 
0}, 
I, 
a constant; 
! second, we 
1) 
6 2 )\’ 
d hence the 
l x = 0, has 
a + d = ol + B, 
Considering now the functional equations, suppose for the moment that g is = 0; 
the two equations may be satisfied by assuming 
cj) (x) = h 
<î> ( x ) = — h 
+ ■ 
c 0 « + d 0 Cj« + dj 
+ ... \ L, 
[«!«+/?! 0£ 2 «+/3< 
We in fact, from the foregoing relations, at once obtain 
a- a- 
,9 
= h 
c — X ' c — X 
+ 
a + A a 2 x + /3 2 " 
6 2 . 6 2 7 
<l> = - /i 
c — « C — X 
c 1 x + d 7 "h c 2 « + d 2 " ” j 
« et)" 
+ 
a 2 6 2 P 
M. 
To satisfy the first equation we must have 31 = aL; viz. this being so, the equation 
becomes 
, b 2 / b 2 \ aLh 
acf)X H $ 
c — x \c —xj c 0 x + d 0 ’ 
or, since c 0 « + d 0 =l, the equation will be satisfied if only aL = 1, whence also 31=1. 
And the second equation will be satisfied if only ^ — bM; viz. substituting for L, 31 
their value, we find co = cib. 
Supposing, in like manner, that h = 0, g retaining its proper value, we 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 
a<f> (x) + 
b 2 
$ 
b 2 
a- 
</> 
c — x \c — X. 
c — x \c — x. 
a 2 
= h, 
viz. the solution is 
4>(x) 
= h -\ 
ab 
h 
a |c 0 « + d 0 Cj« + dj 
+ b<&(x)=g; 
ab 
} 9 |a x « + bj *** a 2 « + b 2 
(ab) 2 
+ .. 
$ (x) = — h 
ab 
+ 
(ab) 2 
+ ...y+i 
+ 
ab 
+ ... 
|a!« +¡3 1 1 a 2 «+/3 2 "’j 1 6}y 0 « + S 0 ' 7!« + Si j 
We have a general solution containing an arbitrary constant P by adding to the 
foregoing values for (f)x a term 
_ Pb(a-b) 
Va 2 (c — x) —x (c 2 — b 2 — cx) 
_ Pa (b — a) 
Vb 2 (c — x) — x (c 2 — a? — cx) ’ 
and for <3>« a term