PROEESSOB CAYLEY ON PREPOTENTIALS. 
749 
* 
so that from the term in a 2 we have 
/ 2 + S 
B / 2 
C / 2 
>+fl.^+r" / 2 +s.a 2 + 
r = 0 
or, what is the same thing, 
A|-2 2 -3-Apj . . ■++*— RT+'O} -b JA . . 
with the like equations from /3 2 . . . y 2 ; and from the constant term we have 
r+o 
4-B— . . .+^“+-*0=0. 
^ o 2 + 0 ^A 2 + 0 
118. Multiplying this last by/ 2 , and adding it to the first, we obtain 
a{- 2 2 -2 D= o : 
fiz. putting for shortness this is 
A{2^+2 + Q+M/ 2 + 5)}+«/ 2 D=0, 
B{ 2 ^+ 24 - 0 +|x(/ + 0)}+*/D=O, 
and similarly 
C|2^+2 + Q+i47i 2 +Q)}+^ 2 D=0, 
and to these we join the foregoing equation 
/2+g+^S • • -+FT0 — «D=0. 
Eliminating A, B . . . C, D we have an equation which determines z as a function of Q ; 
and the equations then determine the ratios of A, B . . . C, D, so that these quantities 
will be given as determinate multiples of an arbitrary quantity M. The equation for * 
is in fact 
P _ L f 
(/ 2 + 0){2 ? + 2 + O + i «(/ 2 + 0)l + (/ + 0){2 ? + 2 + O + i «(/ + 0)} ' ' 
A 2 
(A 2 + 0){2 ? + 2+O + i«(A 2 +0)} 
and the values of A, B . . C, D are then 
M / 2 Mg 2 
MA 2 
+ 1=0 
M 
2g + 2+O + ix(/ 2 + 0)’ 2g + 2 + fl + ijc(/ + 0)’ 2g + 2 + 0 + ±x(A 2 + < 
values which seem to be dependent on d: if they were so, it would be fatal to the success 
of the process ; but they are really independent of 0. 
119. That they are independent of 0 depends on the theorem that we have 
( 2g + 2 + O)* 0 
" 2q + 2- 
5 g 2 
