272 
MR. BOOLE ON A GENERAL METHOD IN ANALYSIS. 
therefore 
and 
/ d \ n ~ 1 tr+n— 1 
u=f \Jt) i»(i-*) m ’ 
1 / d\ n ~ l t r+n ~ l 
T(n)f* 
l\dt 
(1 ~t) n 
( 79 .) 
wherein t — v^, a formula of great simplicity. 
Hence then we find 
,!T dx cos rx 
S*<7r 
J b (1 — 
(1 — 2 vcosa’ + v 2 )' 4 ' 
T(ri)l 
n— 1 fr+n— 1 
(i -t) n 
(80.) 
Legendre has, I believe, considered the above definite integral, but I am not ac- 
quainted with the results of his analysis. 
The following expression for the value V of the definite integral 
dx cos rx 
/>5T 
do (I —5 
( 1 — 2v x cos x + Vj 2 y( 1 — 2v 2 cos x + v 2 2 ) m ...g factors 
is remarkable for its symmetry, 
( d \ l ~ l 
/ d \ m_1 
/ /— 1/ m— 1 
\dtj 
\dtj 
IMS'*'" 
+ t .(» 
V+« -1 
+&c.] (81.) 
wherein, after effecting the differentiations, we must change t } into v x 2 , £ 2 into r., 2 , &c. 
observing that 
i W j l W /"i Wj. 
Tr— 
^2_\/^l_ ^3_\ /^1 
vi WVi * 3 / vi w 
and so on for the rest. 
Ex. 2. To express by a partial differential equation the fundamental properties of 
the definite multiple integral 
u=ff..dx l dx 2 ..dx n (p(a l —x 1 , a 2 —x 2 ..a n —x n ) (82.) 
the integrations extending to all real values of x x x 2 ..x n , subject to the inequality 
ry 2 /yi 2 ry 2 
1 _L 2 i | 
2~r j 2---"r /. 2C 1 - 
/«! He, H n 
We may consider the above integral as derived from the more general one, 
u = ff. . dx l dx 2 . . dx n <p ( a x — a 2 — x 2 t. ..a n — xj ) , 
by the assumption t— 1 . If we expand ^(flq — a 2 — x 2 t..a n — x n t ) in ascending powers 
of t, and integrate within the proposed limits by the aid of Dirichlet’s theorem, we 
find 
U— 24:..2pM{n + 2).\n + 2p)^ 1>t, ’^ a ^’‘ %a n)- 
r(fi p - 
( 83 .) 
V 
wherein 
A — 'b daf' 2 ,ln ‘ 
2 rfflg 
* Cambridge Mathematical Journal, No. XIV. p. 69. 
