784 PEOFESSOK BOOLE ON THE COMPAEISON OF TEANSCENDEXTS, WITH 
by some formulse of Mr. Cayley’s connected with the reduction of the integrals which 
occur in certain problems of this class. I have not, however, attempted a verifi- 
cation. 
34, Although the list which I have given of results obtainable by other methods 
might be increased, it is still only in comparatively rare instances that the means of inde- 
pendent verification present themselves. We might by transformations such as Cauchy 
has employed, verify the theorem 
dx co%m( x—-\ 
1 + 
but it would not be easy by any such process to verify the theorem 
dx cosl m 
i(x ^\\ 
V ^ ^n— K/ j . 
l + (x—^ 
and being any positive quantities, and Xj, Xj, &c. any real quantities whatever. 
I shall not, however, dwell any longer upon special results, but shall briefly state some of 
the general consequences which flow from the application of the primary theorem (1.). 
1st. The evaluation of any definite integral 
in which <p(x) is a rational and hitegral function ofx, is redudhle to that of the definite 
integral 
'^(v)f(v)dv, (1.) 
%/ — 00 
in which \p(v) is a rational and integral function of an 07'der not higher than the order 
of <p(x). 
For, by the general theorem. 
J dx(p{x)f{x- 
cc A j 00 Ag ^ 
i; — :r-J- 
x — X 
+ ■ 
I X — X„ 
1 
= \ dvf{v)C^ 
(p{x) 
x-v- 
^ X — 
since (p{x) is integral. If we develope the fractions -, &c. in descending 
«,t*“Aj .f— A^ 
powers of x, we shall have 
<p{^) 
x-v- 
X ^1 X — 
x-v- 
X X^’ 
