768 
PROFESS OE CAYLEY ON PREPOTENTIALS. 
the equation 
i_ _ 6 __ e n 
0+/ 2 '-- 0 + A 2 0+F 
has a positive root differing from zero, which may be represented by the same letter 0 ; 
but if 
« 2 c 2 e 2 1 
/ 2 • • ' + A 2 + A* <1 » 
then the positive root of the original equation becomes =0; viz. as l gradually dimi- 
nishes to zero the positive root d also diminishes, and becomes ultimately zero. 
Hence writing 1=0, we have 
dS 
{{a-fxf . . . +{c-hzf + (e-kwy}i* 
or, what is the same thing, 
dx . . .dz 
™s- 
+ w{(a— <r) 2 . . . +(c— z) z +(e + Aw) 2 } 2 ® 
wr.< 
dt 1- 
Q now denoting the positive root of the equation 
q/ l gl 
1 ~Q+p‘ • • “0 + F - 0 + F =0, 
t+li 1 . t+Tc 2 )^, 
^•••+^+F>lor<l. 
a 2 e 2 
In the case • +^<1? the inferior limit being then 0, this is in fact Jacobi’s 
theorem (Crelle, t. xii. p. 69, 1834) ; but Jacobi does not consider the general case where 
l is not =0, nor does he give explicitly the formula in the other case 
n a 2 , c 2 e 2 
1=0, . . . + A2 +Ii>- L - 
A 2 1 A 2 ' 
150. Suppose Jc= 0, e being in the first instance not =0, then the former alternative 
holds good ; and observing, in regard to the form which contains +w in the denomi- 
nator, that we can now take account of the two values by simply multiplying by 2, we 
have 
dS 2 C dx ... dz 
I {{a-fxf.. . + (c-hzf + e*\ is ’ /.../il|M 2 ... + M 2 + « s F ! 
(w on the right-hand side denoting ' y/i—^ 3 ... —p, and the limiting equation being 
a? 2 z 2 
j 2 • . • + p=l), each 
