[607 
positive 
^,-1: 
v-9 
point 
= -q. 
)n the 
(1848), 
>0 (the 
ìemoir), 
ications 
5—803, 
was in 
of in 
and i), 
607] 
A MEMOIR ON PREPOTENTIALS. 
421 
viz. the right-hand side is here equal to the left-hand side or is = 0, according as 
0$* 
7^+...+t- <1 or >1. V is consequently obtained by multiplying the right-hand side 
by dx... dz and integrating from — oo to + oo for each variable. 
Hence, changing the order of the integration, 
V = —^——\ iff du, dv dr v* s+q cbu. il, 
7T T (4a 4- a) Jo Jo Jo 
where 
fi = J dx ... dz cos - e 2 T — j-% - • . • —jr 2 + t {(a — x) 2 + ... 4 (c — z) 2 } )v + % (|s + q) ir\. 
Now 
if 
jr 2 + t (a — x) 2 = £ 2 + 1+ p T 
ra? z 2 y, 1 + k 2 T tc 
, .., n + r (c-z)- = — rr -? + 
h 2 
1 + h 2 T ’ 
f = x — 
f 2 Td 
1 +f 2 T ’ 
h 2 TC 
1 4- Ji 2 t ' 
158. Substituting, and integrating with respect to £, between the limits — go 
+ qo , we have 
n = 
(/... A) 774 s 
{(1 +/V) ... (1 +A 2 r)}M 8 
cos w — e-T — 
1 . 
a-T 
(TT 
1 + _/ 2 t ” * 1 + 1i 2 t 
)^+2^7rJ; 
or, what is the same thing, writing - in place of r, this is 
v 
il = 
that is, writing 
we have 
(f ... A) 7T^ S $ 
{(/*+ $)••• (A 2 4-£)}*^ s 
COS -u m — 
/ 2 -M *** A 2 4- £ 
-f)«+ !«*■}; 
a 2 c 2 e 2 
<r= /íTí + ••• +IГT ' + ■'• 
TT^- 1 (/... A) f 1 
r(*« + g) 
m c 
0 
dii dv d£ 
li 2 +t t 
t~ q ~ 1 v ? cos {(ii — cr) v 4- ^q7r] (f>u 
{(t +/»)...(* + A*»i ; 
or, writing 7r^' _1 = - (r^-)*, this is 
= f dt. t~ q ~ x {(£ + / 2 ) ... (t + A 2 )}“* -if dudv. v q cos {(w — a) v + ^g"7r} </m. 
I(is + O)jo TT J oJ 0 
r (hs + q) 
159. Boole writes 
1 ri r°° / d \ q 
- J | du dv v q cos {(u — <t) v 4- %qir] <j>u = </> (<r); 
viz. starting from Fourier’s theorem, 
— f j dudv cos(u — a) v . <fiu = <j> (<r), 
7TJ o J 0