758 „ 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
We have 
where 
r(^-i) / dW dw" \ 
’~“4(r|)* +I \ da' + da" )• 
dW' 
da' z 
(s 2) (««... +« 8 )M _ (s _ 2)M dW" n 
— at the suriace ; -^-=U, 
and therefore 
(**+...**)* /< 
' (s-2)r(i s -i)M 
4 ( r i) s+1 / s 
(viz. g is constant) ; 
or, what is the same thing, writing M = (s— l C ’ w ^ ence S-ty and writing also 
a 2 . . . + c 2 =z 2 , we have 
dfdS 
(a—x ) 2 . . . + (c—z) 2 + (e— wYY 
4(T X)s + lfs- n f 1 . , ^ 
= ( g _2)r4s— i) tf=* for an extenor P omt ;i >h 
4(ri), +] / s -. 8 / i ^ x ^ 
= (s_2)r( l^ -i) /^ for interior point *</. 
132. This is right; but we can without difficulty bring it to coincide with the result 
obtained for the (s-f-l)dimensional sphere with only s — 1 in place of s ; we may in 
fact, by a single integration, pass from the cylinder x 2 . . . -\-z 2 =f 2 to the s-dimensional 
sphere or circle x 2 . . . -\-z 2 f 2 , which is the base of this cylinder. Writing first dS=d'%dw, 
where (72 refers to the s variables [x ... z) and the sphere x 2 . . . -\-z 2 =f 2 ; or using now 
dS in this sense, then in place of the original d&> we have dSdw : and the limits of w 
being co , — co , then in place of e—w we may write simply w. This being so, and 
putting for shortness (a— x) 2 . . . -\-{c — z) 2 —A 2 , the integral is 
J-. J (A»+«*)K- 
and we have without difficulty 
dco 
r 
i r£r$(« 
.. (A*+*d*) 1( *“ 1) A- 2 Fi(s-1) • 
[To prove it write w — A tan 3, then the integral is in the first place converted into 
A s_ - 
cos s 3 QdO, which, putting cos 6=\/ x and therefore sin Q=*yi— x, becomes 
=A f vf-v-'dx, 
which has the value in question.] 
Hence replacing A by its value we have 
A r iJ.5-2) C SfdS 
F}( s — !) J I (a— x ) 2 . . . + (c- 
4 ^T(i).f- 1 8/ f 
^) 2 P (S “ 2) (s -l)\(a*...+c?f s - 2) f s ~ 2 
