PROFESSOR CAYLEY ON PREPOTENTIALS. 
751 
Substituting for A, B, . . . C their values, this last becomes 
JD 
2 q + \ 
XnD 
viz. this is 
2 0 ) 
2q + 2 + ik 0 Q\2q+2 + i* o f 2 ~f + 0$ ' ' ' ~ 2q + 2-^ 0 &\2q + 2+^~ 
- 2? +2-i^ {22 + 2 + Q}=0; 
■ ■ ■ +{2 ? +1^^-FT8} +2 2+ 2+Q=0; 
or substituting for O its value, and dividing out by 2 ^+ 2 , we have 
v+ 0 ; 
°q + 2 + U 0 f 2 ^2q + 2 + ±x 0 g‘ 
2 * ’ • + 2j + 2 + ±x 0 A 2 + 1 — 
the equation for the determination of z 0 . 
122. The equation for z 0 is of the order s; there are consequently s functions of the 
form in question, and each of the terms a 2 , 0 2 , . . . y 2 can be expressed as a linear func- 
tion of these. It thus appears that any quadric function of a, (3, ... y can be expressed 
as a sum of Greenian functions ; viz. the form is 
A 
+Ba-{-&c. 
-{- C «3 -f- &c. 
[D ,/ l/ 2 * 2 - W . . _i_\ 
^ ^ 2q + 2 + A * 0 '/ 2 ^ 2 ? + 2 + ^ * ' • 2q + 2 + x<) 
+D"( „ „ „ ) 
(s lines), 
viz. the terms multiplied by D', D", &c. respectively are those answering to the roots 
z 0 ', z ", ... of the equation in z 0 . 
The general conclusion is that any rational and integral function of a, 0, ... y can be 
expressed as a sum of Greenian functions. 
123. We have next to integrate the^ equation 
40 
^( 2 2+2 +j^r Q +^r Q . . . -zQ= 0 . 
Suppose z— 0, a particular solution is 0 = 1 ; 
7 2 +a 
7 + < 
. . — 72 ^^ , a particular solution is — _ V/' + ^ . — . 
in fact, omitting the constant denominator, or writing 0 = v //"+^, ancl therefore 
d© l l 
~ 2 \// 2 + ~~ 4(/ 2 + 0 ) i ’ 
