798 PEOFESSOE BOOLE OX THE COMPAEISOX OF TEAXSCEXDEXTS, TTETH 
’SMience, x having only one value in terms of v, 
f{v)dv=e[J{x)']^, 
and integrating, 
Sfiv)(lv= 0 [/(^)] {log (v-x) } , 
■<p(x) being an arbitrary function of x. It is easily seen that ©[/(•T)]'4'(>y) may be repre- 
sented by C, whence 
jf(v)d'v=e[f(x)] log (v-x)-i-C (6.) 
Cauchy has very extensively employed the Calculus of Residues in the evaluation of 
definite integrals taken between the limits 0 and co. These apphcations are among the 
most valuable portions of his writings. They have no connexion, however, with the 
researches of this paper, and I have not even examined whether they would be in any 
degree generalized by the adoption of the symbol 0. 
Note B. 
On the Interpretation of the Formulae for the Evaluation of Multiple Integrals. 
The three principal formulae, (4.) art. 38, (1.) art. 41, and (5.) aid. 42, evidently pos- 
sess a common type. In each of them we recognize under the sign f, a function /(<r) 
which may be discontinuous within the limits of integration, and which is at the same 
time subject to an operation of general differentiation. This is a comhmation which is 
at least unusual in analysis. I purpose to consider here some of the questions of inter- 
pretation which it suggests. To some extent, indeed, these questions have been con- 
sidered in my previous memoir, already referred to ; but one of the most important of 
them, the effect of discontinuity in the function /’(c) upon the integral in which it is 
involved, admits of being presented in a more satisfactory light. I do not propose to 
enter upon a complete investigation of the latter question, but only to examine one or 
two special and well-marked cases, in the hope of directing the attention of others to the 
subject. 
When there is but one variable, and the index of differentiation is 0, the formulae 
reduce in effect to ordinary integral transformations. And it is quite worthy of obser- 
vation, that in this way the formula (4.), Art. 38, leads to kno’s^m modular transformations 
of the elliptic functions. 
Thus if we make w=l. 
7(=1, i=i, 
ffj = 0 and di'op the suffix 
from x^ and l^, we have 
where a=s — ^ andy'(ff) 
to saying that 
I dxfllx) Ids . 
j(Tf^=j7/W. 
vanishes when a transcends the limits of lx. 
.... ( 1 .) 
But this amounts 
dxf{lx) 
