Mr. Boore on a certain Multiple Definite Integral. 141 
The idea of the employment of discontinuous functions seems also to have inde- 
pendently occurred to Mr. Ellis,* who makes use of Fourier’s theorem, and is 
thus led to some elegant results, which M. Dirichlet’s process would fail to dis- 
cover. We believe it may ultimately be found that all definite multiple integrals 
of which the finite evaluation is possible, may be resolved into distinct classes, 
each dependent on some primary discontinuous function. In the example sub- 
joined we shall make use of the theorem, 
= aS, 5, dade dw cos } (a—x)o—teo +n hor f @) sa) a) 
in which f(2) may be discontinuous; and the first sign of integration in the 
second member is to be used in the same way as the corresponding one in 
Fourier’s theorem. The two remaining signs are used in the “ limiting’’ sense, 
according to which (ydvp(v), SSdw@(w), are considered as the limits of 
{0 dve"""O(v), ( dwe"”(w) 5 k being a positive quantity approximating to 0. 
2. It is required to evaluate the definite multiple integral 
a ee 
Ae Wee CAH a ae Siti) (2) 
[(4,— x) a (@,— r,)’. J’ 
the number of the variables being », and the integrations limited by the con- 
dition 
pe 22 das ; 
We may remark that the value of this integral cannot be deduced from the for- 
mule of M. Dirichlet and others, although in some cases such generalizations 
may be effected. 
2 - 2 
ee rie \ tp ee : : : F 
Since hi ~ he .. 1s a discontinuous function lying between 0 and 1, we have 
by (1), 
Se + = f :) 1 1p ax ay’ 
i te = - dadi i-1 
Garay ew) NS, S OE SOE TG: 4 
* Cambridge Mathematical Journal, Nos. XIX. XX, XXI. 
