PROFESSOR CAYLEY ON PREPOTENTIALS. 
769 
'<-(*+/’•• -«+m 
w C * c • • 
where 0 is here the positive root of the equation 1—f^p • • • ~f^p~ y=0, which is the 
formula referred to at the beginning of the present Annex. We may in the formula 
write 0 = 0 , thus obtaining; the theorem under two different forms for the cases 
p • • • > 1 an d < 1 respectively. 
Annex X. Methods of Lejeune-Dirichlet and Boole. — Nos. 151 to 162. 
151. The notion that the density § is a discontinuous function vanishing for points 
outside the attracting mass has been made use of in a different manner by Lejeune- 
Dirichlet (1839) and Boole (1857) : viz. supposing that g has a given value f{x .. . z ) 
within a given closed surface S and is =0 outside the surface, these geometers in the 
expression of a potential or prepotential integral replace g by a definite integral which 
possesses the discontinuity in question, viz. it is =f{x . . . z) for points inside the surface 
and =0 for points outside the surface ; and then in the potential or prepotential integral 
they extend the integration over the whole of infinite space, thus getting rid of the 
equation of the surface as a limiting equation for the multiple integral. 
152. Lejeune-Dirichlet’s paper “ Sur une nouvelle methode pour la determination 
des integrates multiples ” is published in ‘ Comptes Rendus,’ t. viii. pp. 155-160 (1839), 
and Liouv. t. iv. pp. 164-168 (same year). The process is applied to the form 
1 d dxdydz 
da}{ (a-x)* + {b-yf+ (c-*)*}**-” 
over the ellipsoid ^+^+^2=1 ; but it would be equally applicable to the triple inte- 
gral itself, or say to the s-tuple integral 
C dx . . .dz 
or, indeed, to 
(c-*y 
dx ... dz 
{ {a— x) 2 . . . + (c— z) 2 + e 2 l 
li»+s 
over the ellipsoid jr 2 . . . +p=l ; but it may be as well to attend to the first form, 
more resembling that considered by the author. 
153 . Since z. 1 cos \<p dq> is =1 or 0, according as X is < 1 or > 1, it follows that 
^ 0 T 
the integral is equal to the real part of the following expression, 
*J 0 ^ 9 ) Ua-x)*... 
\{a-x)z . . . +(c - s) 2 ? 
