A MEMOIR ON PREPOTENTIALS. 
416 
[607 
provisional assumption. Or, writing for greater convenience 6 to denote the positive 
quantity — L, that is, taking 6 to be the positive root of the equation 
we have 
I \(a - 
1 - 
dS 
a- 
e+p 
e+h 2 e+k 2 e 
e2 l -=o 
{(a —fx) 2 + ... + (c — hzf + (e — kw) 2 + lp s 
=^fdt 
1 2 S J 0 
tsj••• (t + h 2 )(t + k 2 )(l - 
l 2 \ ’ 
t+P 
t+h 2 t + k 2 t 
or, what is the same thing, we have 
If dx ... dz 
P7hJ T 
/• 
± w {(a — x) 2 + ... + (c — z) 2 + (e + kw) 2 + Z 2 }* s 
f 
dt 1 
[t(t+f 2 )...(t + h 2 )(t + k 2 )}~K 
t+r 
t + h 2 t+k 2 
j 
where on the left-hand side w now denotes ^ 1 — —... and the limiting equation 
. X 2 z 2 _ 
“75 + - + r* =1 - 
149. Suppose 1 = 0: then, if 
c e 
the equation 
f 2+ '" + h 2 + k 2>1. 
e- 
6+f 2 *” e + h 2 e + te 
= o 
has a positive root differing from zero, which may be represented by the same letter 6; 
but if 
a 2 c 2 e 2 
f‘ Jr - + h* + ¥ <1 ’ 
then the positive root of the original equation becomes = 0; viz. as l gradually 
diminishes to zero, the positive root 9 also diminishes and becomes ultimately zero. 
Hence, writing 1=0, we have 
I\(a-fxf + ... 
dS 
{(a — fxf + ... + (c — hz) 2 + {e — &w) 2 p ’ 
or, what is the same thing, 
dx... dz 
2 my 
ns 
/*00 
J e 
f... h J ± w {(a — x) 2 + ... + (c — z) 2 + (e + kw) 2 )^ s * 
a 2 c 2 e 2 N ~ * 
dt 1 — 
t+P 
t + h 2 t + k 2 
) [t{t+p) ,..(t + h 2 )(t + k 2 )\ *,