770 
PKOPESSOB CAYLEY ON PEEPOTENTIALS. 
where the integrations in regard to x ... z are .now to be extended from —go to + co for 
each variable, A further transformation is necessary : since 
1 l 
-=Y r e ~ r7!i j d-Aj . 1 e"*, a positive and r positive and <1, 
writing herein (a—x) 2 . . .-\-(c—zf for <r, and \s-\-q for r, we have 
.. •■■ ■ 1 1 rlxl 
and the value is thus 
O 
g-Qs+q) 
J dx... dz , 
nT^sH-g) 
where the integral in regard to the variables (x . . . z) is 
=#«*- + ^jdx^ (+ + ^> + 3 a **i- . . . j dzA o+ty-wi ; 
= eli V£ 
-e f^+v, 
f*t+9 
and the like for the other integrals up to the 2 -integral. The resulting value is thus 
sin <p 
‘7lT(^S + 9') 1 
d-Aj ■A s+2_1 e^ 4 G+/^' , ' + p+fc 2 '/') — , 
+/**... ?+***’ 
which, putting therein 4'=^, ^ dt, is 
eft 
/-2- 1 
s/p + t...h* + t J 0 
j e^^f 2+t ' ’ ’ + *‘+t) sin i p . <jd 2_1 <7<p. 
’ n 
154. But we have to consider only the real part of this expression; viz. writing for 
a' 2 c 2 
shortness a=j ^~ t . . . we require the real part of 
e ~ iq ” J e icr<f . <p 2_1 sin <p dcp. 
Writing here for sin © its exponential value^-. (e i<p —e i<p ), and using the formula 
e~ qni j d<p . <p q ~ l . (o- positive), 
and the like one 
elni j o d( P • <P 2-1 ^ (<r negative) 
(in which formulse q must be positive and less than 1), we see that the real part in 
question is =0, or is 
Tg sin (g + l)7r 7r 1 
2(1— (r)* 5 ~2Y(\—q) (1 — tr)«’ 
according as cr > 1 or <r < 1. 
