746 
PEOFESSOE CAYLEY ON PEEPOTENTIALS. 
112. If 77i=l, we have 
1 — 
f % " h* 
dx . . .dz 
\[a-xf... +(c-z)* + e‘ 2 }i s+ « 
(rj)T(2+ g ) 
r &+q) 
which, differentiated in respect to a, gives the (a— ^formula ; hence conversely, assuming 
the a—x, . . .c—z, e formula, we obtain by integration the last preceding formula to a 
constant pres, viz. we thereby obtain the multiple integral =C+ right-hand function, 
where C is independent of (a ... c, e) ; and by taking these all infinite, observing that 
then d—co , the two integrals each vanish, and we obtain C=0. 
In particular s= 3, g= — 1, then 
Jt 
dx dij dz 
(a— x) 2 + {b— y)‘ 2 + (c— z) 2 + e 2 j 
p *+*’)•*» 
which, putting therein e=0, gives the potential of an ellipsoid for the cases of an 
exterior point and an interior point respectively. 
Annex Y. Green’s Integration of the Prepotential Equation 
(* 
\ do? ’ 
P.P 2 g+1 d 
' dc 2 ' de 2 ' e 
* V: 
:0.— Nos. 113 to 128. 
113. In the present Annex I in part reproduce Green’s process for the integration of 
this equation by means of a series of functions analogous to Laplace’s Functions, and 
which may be termed “ Greenians ” (see his Memoir on the Attraction of Ellipsoids, 
referred to above) ; each such function gives rise to a Prepotential Integral. 
Green shows, by a complicated and difficult piece of general reasoning, that there 
exist solutions of the form V =0<p (see post, No. 116), where <p is a function of the s 
new variables a, 3 ... y without 6, such that \7(p=zcp, z being a function of Q only ; these 
functions <p of the variables a, 3 ... y are in fact the Greenian Functions in question. 
The function of the order 0 is <p= 1 ; those of the order 1 are <p=a, <p=/3 . . . <p=y ; 
those of the order 2 are <p=a(3, See., and s-functions each of the form 
2"{ A a? -fi B3 2 • • • + Cy 2 } + D. 
The existence of the functions just referred to other than the s-functions involving the 
squares of the variables is obvious enough ; the difficulty first arises in regard to these 
s-functions ; and the actual development of them appears to me important by reason of 
the light which is thereby thrown upon the general theory. This I accomplish in the 
present Annex ; and I determine by Green’s process the corresponding prepotential 
integrals. I do not go into the question of the Greenian Functions of orders superior 
to the second. 
114. I write for greater clearness (a, b . . .c, e ) instead of (a ... c, e) to denote the 
series of (s + 1) variables; viz. ( a,b...c ) will denote a series of s variables; corre- 
sponding to these we have the semiaxes (f, g .. . h), and the new variables (a, (3 . . . y) ; 
