704 
PROFESSOR CAYLEY ON PEEPOTENTIALS. 
and the resulting value of $ is found to be =0 for values of {cc . . . z) for which 
x 2 . z 2 -j 
dx . . .dz 
{{a-x)\ .. + (c-zf+ e 2 )^ 
™2 ,,2 
the limiting equation being say this is the s-coordinal ellipsoid. It is 
clear that this includes the before-mentioned case of the s-coordinal sphere; but it is, 
on account of the more simple form of the ^-equation, worth while to work out directly 
an example for the sphere. 
46. Three examples are worked out in Annex IV. ; the results are as follows : — 
First, 0 defined for the sphere as above; ^ + 1 positive; 
Jfczz 
J {(a—x ) 2 ...- 4 
dx. . . dz 
,2 li *+2 
-xy...+(c- zy+e*y 
over the sphere x 2 . . . +y 2 =f 2 , 
=S T wfflfft-'-'V+f )-***■ 
This is included in the next-mentioned example for the ellipsoid. 
Secondly, 0 defined for the ellipsoid as above ; g'-j-l positive ; 
f (l-t- 
. — Ts 1 dx . . .dz 
Y—\ ' f 
h 2 ) 
J {{a—x ) 2 . . 
. + {c-z) 2 +e 2 ) is+2 
a? 2 z 2 
over the ellipsoid 
1 
This result is included in the next-mentioned example ; but the proof for the general 
value of m is not directly applicable to the value m= 0 for the case in question. 
Thirdly, 0 and the ellipsoid as above ; y + l positive; m — 0 or positive, and apparently 
in other cases. 
V=f ( 
, 1+ P" 
gt\ 2+« 
\a-xf.. 
. + {c-z) 2 + e 2 \ is+2 
: the ellipsoid as above. 
(ri) s r(i+g+m) 
— rGs+^)r(i+m) v 
d 2 
f+r 
,3 p2\ m 
• • • t+e)-Ht. 
And we have in Annex V. a fourth example ; here Q and the ellipsoid are as above : 
the result involves the Greenian functions. 
