762 
PBOEESSOB CAYLEY ON PBEPOTENTIALS. 
(notation to be presently explained), being a result obtained by Jacobi by a process 
which is in fact the extension to any number of variables of that made use of by 
Gauss in his Memoir ‘ Determinatio attractionis quam exerceret planeta, &c.’ 
(1818). I proceed to develop this theory. 
139. Jacobi’s process has reference to a class of s- tuple integrals (including some of 
those here previously considered) which may be termed “ epispheric ” : viz. considering 
the (s+1) variables (x...z,w) connected by the equation x 2 ...-\-z 2 +w 2 = 1, or say they are 
the coordinates of a point on a (s+l)tuple unit-sphere, then the form is JU^S, where 
dS is the element of the surface of the unit-sphere, and U is any function of the 5+1 
coordinates : the integral is taken to be of the form { . , ^ — — rr-pu, 
and 
then 
Before going further it is convenient to remark that taking as independent variables the 
s coordinates x...z, we have dS = ^' x where w stands for + */l —x 2 ...—z 2 ; we must 
dw ~ 
in obtaining the integral take account of the two values of w, and finally extend the 
integral to the values of x ... z which satisfy x 2 . . .+z 2 < 1. 
If, as is ultimately done, in place of x . . . 
value of d$ is = 7-? where w now stands for + 
we write - 
respectively, then the 
\/ ] 
xd 
7 2 "' 
— ~ ; we must in 
w ' v /*"’ 
finding the value of the integral take account of the two values of w, and finally extend 
the integral to the values of x 
which satisfy^ 
140. The determination of the integral depends upon formulae for the transformation 
of the spherical element <7S, and of the quadric function (x, y . . . z, w, l) 2 . 
First, as regards the spherical element dS ; let the s + 1 variables x, y . . . z, w which 
satisfy x 2 -\-y 2 . .. z 2 -\-w 2 = 1 be regarded as functions of the s independent variables 
then we have 
X, 
y • 
• • *5 
w 
dx 
dy 
dz 
dw 
HP 
dQ ’ 
’ ‘ dP 
HQ 
dx 
dy 
dz 
d 
df’ 
* * dtf 
dtp 
dx 
dy 
dz 
dw 
w 
# * 
■ ■ 
dty 
we 
effect 
on the s+ 
dQdxp . . . iTv}/, = 
d(0 3 <P *) 
d$ d<p . . . d\ p, for shortness. 
w) a transformation 
x,y 
X Y 
’T’ T* 
Z W 
• T > y ’ 
