140 
VIL.—On a certain Multiple Definite Integral. By Grorce Boote, Esq. 
Read 13th April, 1846. 
1. IT has for some time been known that the evaluation of multiple definite 
integrals can in many cases be effected by the employment of discontinuous func- 
tions. M. Lejeune Dirichlet has, in this way, obtained the value of the multiple 
integral, 
On ine rec aa 
Ni cesea acess as 
the limits being given by the inequality, 
He founds his analysis on the properties of the discontinuous function, 
\ do sin  cosrp 
“o p j 
e ee Tv < . s . . ° 
which is equal to = when 7 lies within the limits — 1 and 1, and vanishes when 
r transcends those limits. As a process of integration, M. Dirichlet’s is, perhaps, 
the most remarkable that has ever been published, but it does not appear that 
any account of it has been published in the English language. For his own 
knowledge of it the author of this paper is indebted to an abstract of M. Dirichlet’s 
memoir, with which he was favoured by Mr. Cayley, whose investigations on 
this subject* are scarcely less important, though pursued by a different method. 
* Cambridge Mathematical Journal, Nos. XIV. XV. 
