CEETAIN APPLICATIONS TO THE THEOEY OF DEFINITE INTEOEALS. 7G1 
The denominator of this expression does not contain x—p as a factor. Hence there 
will be no factor of the above description in the denominator of the fraction within tlie 
brackets, and therefore no corresponding development in the performance of 0. Thus 
the only factors which produce any effect are those found in the denominator of F. The 
part of the operation 0 expressed by — Ci is of course unaffected by the nature of the 
function within the brackets. 
On these accounts then the theorem (11.), seeing that its second member indicates the 
SE 
performance of the operation 0 on the function F^, the interpretation of that operation 
being derived solely from the factor F, is reduced to the comparatively simple form 
2F<ii-=0[F]^ (14.) 
And in this form it constitutes the general theorem of transformation which it was 
required to demonstrate. 
Application of the general Theorem of Transformation to the Comparison of 
A Igehraical Transcendents. 
16. In treating of the algebraical transcendents, I shall first exemplify the direct 
application of the general theorem of transformation to the solution of special problems, 
and for this application I shall select by preference examples known and familiar. I 
shall subsequently apply the theorem to the investigation of general formulse from which 
the solution of all special problems may be derived. 
There is no difficulty in the direct application of the theorem to special problems. 
The following directions will meet every case. 
Let 2 j 'K.dx be the expression whose finite value is to be found, the simultaneous values 
of X being determined by an equation 
X=F(^, .. «^), (1.) 
flj, •• (ir being the new variables in terms of which the value of the integral expression 
is to be obtained. The second member, which we shall represent by F, is supposed 
rational with respect to x. Let also the equation (1.), made rational and entire with 
respect to x by reduction to the form 
PdA^A-pxxr 
be represented by E=0. 
Then observing that X does not become infinite when E=0, we have 
( 2 .) 
(3.) 
On performing the operation 0 in the second member, the function under the sign of 
integration will be an exact differential relatively to a^, ..a,., and, being integrated, will 
MDCCCLVII. 5 G 
