750 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
where x 0 is a quantity independent of 0 determined by the equation 
> + < 
lq + 2 + fr o/ 2 2q + 2 + ±x 0 g 
2 • + 
2q + 2 + ±x 0 h* 
+i — o, 
(x 0 is in fact the value of x on writing 0=0), and that, omitting the arbitrary multiplier, 
the values of A, B . . . C, D then are 
/ 2 £ 2 A 2 _1 . 
2q + 2 + \x 0 P' 2q + 2 + \K 0 g*' > ' ' ' 2g + 2 + i^ 0 A 2 ’ x 0 * 
or, what is the same thing, the value of <p is 
- i/ 2 * 2 ■ W , PV . i 
2g + 2 +^ 0 / 2 ' r 2 !? + 2+^ o5 r 2 * - ■^ r 2q + 2 + ^P * 0 * 
120. [To explain the ground of the assumption 
_(2£+2 + % 
/6— 2q + ’ 
observe that, assuming 
2g+ 2 + 0 + |x(/ 2 + 8) _ 2f/ + 2+n + ^(/ + 5) 
2g' + 2 + ix 0 / 2 — 2g + 2+i?f 0 / ’ 
then multiplying out and reducing, we obtain 
i* 0 (22 + 2 + <W-f ) + (2ff+2) • 0 ; 
viz. the equation divides out by the factor g 2 —f 2 , thereby becoming 
x 0 (2q + 2 + Q) - (2$+ 2> + i** o 0= 0, 
that is, it gives for x the foregoing value : hence clearly, x having this value, we obtain 
by symmetry 
2 ? +2+Q+i*[f +0), 2 ? +2 + Q+ W+«)> • • • 2j+2 + Q+A*(7 f + 6), 
proportional to 
2^+2 +ix 0 f, 2q+2+±x 0 g\ . . . 2q+2+ W> 2 ; 
viz. the ratios, not only of A : B, but of A : B . . . : C will be independent of 0.] 
121. To complete the transformation, starting with the foregoing value of x, we have 
so that we have 
2 ? +2+a+Hf +0)=(2 ? +2+fi) . : 
A\2q+2 + iz„f\+* t fT)=0, 
B\2q+2+i^f-}+y.^D=0, 
Cj2 2 +2+|* t A 3 »+Vi ! D=0, 
_ A , B_ , C (2g + 2 + OKP 
/* + 0" 1 V + fi ' ’ ‘ + 2g + 2— — U> 
