790 PEOFESSOE BOOLE OX THE COMPAEISOX OE TEAXSCEXDEXTS, WITH 
Application of the General Theorem to the evaluation of multiple definite integrals. 
37. The form in which multiple definite integrals present themselves in the applica- 
tion of mathematics to natural philosophy, is usually the following. The value of a 
triple integral, 
( 1 .) 
is required to be found, the integration extending, in some instances, to all positive 
values, but more generally to all values whatever of the variables x. y, z which satisfy a 
condition 
■^{x,y,zy^\ ( 2 .) 
The most general method of treating this class of problems is due to M. Lejeuxe 
Dirichlet. It consists in converting <p[x, y^ z) into a discontinuous function which vanishes 
whenever the variables x,y, z transcend the limits assigned m (2.), and which is equal to 
<p{x., y, z) whenever those variables satisfy the above condition. This transformation 
being effected, we are permitted to regard the integrations relatively to x, y. z as inde- 
pendent, and as individually taken between the limits — oo and oc . 
From this circumstance the progress of our knowledge of multiple definite integrals 
must be in some degree coordinate with the extension of oui’ command over single defi- 
nite integrals taken between the hmits — oo and oo . I propose in this section both to 
illustrate, by one or two examples, the theory of multiple integrals as above stated, and 
to show how it gains extension from the theorem of definite integration demonstrated in 
the preceding pages. 
There are several different forms in which the application of the principle of discon- 
tinuity to the evaluation of multiple integrals may be presented. The form which I 
shall adopt in this paper is similar to one originally given by me in the Transactions of 
the Royal Irish Academy (vol. xxi. pt. I), but is more convenient in application. It 
depends essentially upon the employment of Fourier’s theorem, \iz . — 
1 
fix) =- j j da.dv. cos {av—xv)fia). 
If we write the cosine in its exponential form, we have 
fia:)=-^^ J (3.) 
Now by known theorems 
r 
h- ni) Jo 
dw 
~ r(i) J. 
dw (4.) 
Multiply the terms of (3.) by those of (4.) taken in the same order, and, converting the 
exponentials into sines and cosines, we have 
fA) 
V 
~^T(qJ \ \ ^cidvdw cos, \av—xv—tiv-\-~^j'^'~ffi)^ 
the theorem employed in the Irish Transactions. 
. ( 5 .) 
