274 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
Ex. 3. To evaluate the definite multiple integral 
y—ffi dx x dx v .dx n 
) 2 + (« 2 - ^ 2 ) 2 - • + K - 2 } ^ 88 ^ 
n being an odd integer, i any quantity whatever , positive or negative, integral or frac- 
tional, and the integrations extending to all real values of the variables s subject to the 
condition 
xj+x.f.-\-xj<\. 
Here ep(a l a 2 ..a n ) = 2 + +a <zf By induction it is easily shown* that 
(d 2 d 2 d z \ p _ 1 . 2i(2i + 2)...(2i + 2p — 2)(2i + 2 — n)(2i + 4 — w),.(2i 4- 2 p—n) 
yda^'daf”' daV (af + « 2 2 . .. + af) 1 ~ {af+afT+fff^ 
Hence the series (83.) becomes on making (a 1 2 -\-a 2 2 ..-\-a n 2r f=r, 
2 7r 2 ^p='»2i..(2i + 2p — 2)(2i + 2 — n)..(2i + 2p—ri) _ 2d 
~'Zp=o 7rT-Kzz?—r^r 7—r-xzr\ r , — 
'(jY‘ 
2A..2pn(n + 2 )..{n + 2 p) 
2tt 
1 
rl !L\.,r'"' u -* r 
-2p 
27 
11 r 1 . * . (2p + 2) [2p + n + 2) 
the law of the series being u_ 2/ ,= + 2i ) { 2 p + 2i- n+ 2) “-»i>-» ; 
or putting —2 p—m. 
u 
(m— 2){m — n — 2) 
— 0 . 
(?« — 2i){m — 2i + n— 2) m ~ 2 ' 
And as the series extends to all the values of u m which can be formed in subjection 
to the law, we have 
u- 
(D-2)(D-»-2)_ 
{D-2i)(D-2i + n— 2) 
To integrate this equation, assume 
(D — n — 1)(D— n— 2) 
& u u—Q. 
Then by (XIV.) 
(D — 2i + n— 2)(D — 2i-\-n— 3) 
Z 16 V = 0. 
(89.) 
(90.) 
(D-2)(D-2i + »-3) 
u ~ (D-«-i)(D-2i) V ’ 
= (D— 2 )..(D— w+l)(D— 2 /+w— 3)..(D— 2 /+ 2 )v, 
b v ’ 
= (D-2)V : / 2 (D-2/+ W -3)^ 
Yi — ] I 71— 1 
wherein the index — 5— / 2 denotes the product of — 5 — factors, decreasing by a 
mon difference 2 . 
The equation (90.), treated by the method of Prop. 1 , C. § 1 , gives 
V =\( V \+Vi) 
a com- 
v 
D— n— 1 . D— n— 1 . 
i _r D-2i + w-2 sy l — °’ V 'i~ D-2i + n-2 eV * — (K 
* Cambridge Mathematical Journal, No. XIV. p. 63, 
