PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
759 
that is 
Ji 
SfdS 4%'- s f s ~ l if ft 1 
{a-xY . . . + (c-zY\* s - 2) -(s-2)m°-2)\(a*. .. + c*f s ~ 2) 01 
1 1 
” r is \ (#...+<*)*-*> 01 f~ 
viz. this is the formula for the sphere with s — 1 instead of s. 
Annex YII. Example of Theorem D. — Nos. 133 & 134. 
133. The example relates to the (s+l)dimensional sphere x 2 . . . -\-z 2 -\-w 2 =-f 2 . 
Instead of at once assuming for V a form satisfying the proper conditions as to conti- 
nuity, we assume a form with indeterminate coefficients, and make it satisfy the con- 
ditions in question. Write 
V=— 2 for a 2 ... -\-c 2 + e 2 >f 2 ; 
[a? . . . + c 2 + e 2 ) 2 2 J 
=A [a 2 . . . +c 2 -f-<? 2 )+ B for a 2 . . . + c 2 + e 2 <f 2 : 
In order that the two values may be equal at the surface, we must have 
_pi=A/ 2 +B, 
dV 
and in order that the derived functions &c. may be equal, we must have 
— (s— l)aM 
f s 
=2A«, &c. 
viz. these are all satisfied if only - -^-,^ =2A. 
We have thus the values of A and B, or the exterior potential being as above 
M 
the value of the interior potential must be 
=^fa+t)-(»-i). a8 -^ 8+e8 
The corresponding values of W are of course 
M M ( n , fl « 2 . . . + 2 2 + w 2 ) 
(*...+•.+-)•* and M (is+ t)-(f-w) 7 }’ 
and we thence find 
^=0 if of ... -\-z" -\-w 2 
rfr-p , i\> M 
§ ~ 4 (fi) s+1 ^ 4 v 2 S 2A2 S +3/f ys+u — (r^) 4 '+l / s + 1 
if x 2 . . . -\-z 2 -\-w 2 <f\ 
