794 PEOFESSOE BOOLE ON THE C0:MPAEIS0N OF TEANSCENHENTS. ^TH 
wh erein <7 = + 2 ( /? + m\sf{l\a\ +m\h\s^f-\ h 2 (/^ + Tlfll 
f{a) being supposed to vanish when a transcends the limits 0 and 1. This theorem 
admits of the same kind of generahzation as the preceding one. It was communicated 
by me, some years ago, to Mr. Cayley, and pubhshed by him, at my deshre, in Liou- 
ville’s ‘Journal,’ vol. xiii. 
If in the above theorem we make a^ = ^, .. a„=0, we have an expression for the value 
of the multiple integral 
J . . dx^dx<i . . dXn 
+ m 
^ 2/ 2bl 
+ mn\Xn+ — 
xz 
the integrations being extended through the mass of the ellipsoid whose equation is 
l\x\-\ \-llx\=\. 
The following example, originally pubhshed by me, but without any intimation of its 
possible extension by means of the general theorem of de fini te integration, in the 
memoir already referred to in the ‘ Transactions of the Royal Irish Academy,’ is of great 
practical importance. 
42. Example 3rd. Required the value of the multiple integral 
V =]*. . dxidx^. . dXn 
, ^2 I 
■ (1.) 
{h^+{a^—x^Y + {a2-x^^.. + {a„—x„Yy 
the integrations extending to all values of the variables which satisfy the condition 
2 2 
( 2 .) 
Here V will be found by integrating with respect to a.’i, x^, ..x^ between the limits 
— CO and oo the expression 
Hence changing, as before, the order of integration. 
V= 
where 
Now 
if we make 
’’■r(f) 
Otio Jo 
dadvds v's' |/’(a)T, 
T=y -^^nCos + 
f,+s{a,-x;f= l±^yl+^ 
1+hls 
