776 PEOFESSOE BOOLE ON THE COMPAEISON OF TEANSCENDENTS, ^ITH 
Now as a rational function of w may be represented under the fonn q, where 
P and Q are polynomials in x not invohing v. The equation of transfonnation then 
becomes 
P 
and in its rational and integral form Qv — P=0. 
Hence by the general theorem of transformation, 
2f(v)<pdx=e[/(v)<p]^ log (Qv-P)=e[f(v)fj 
Now the factor /(v), inasmuch as it does not contain x, in nowise affects the interpre- 
tation of 0. Hence it may be removed from wit hi n the brackets and the equation 
written in the form 
2/(«)j>&=/WeM^=/(«)eM 
on replacing ^ by >4/. 
Hence, replacing the first member by the expression for which it is an equivalent, and 
attaching the sign of integration, 
( 3 -) 
To this result we may also give the form 
(4.) 
as the symbol 0 and the function <p are both independent of the variable v. And this 
would in fact be the best form if we could effect generally the integration in the second 
member. For the applications to which we shall proceed (3.), is, however, the fonn to 
be preferred. 
We may express the results which have been arrived at in the following general 
theorem. 
Theorem. — If ^ and are rational functions of x, and if the sinmltaneous values of 
X in the integrals included in the expression 2j’<^(4')dx are roots of an equation 
\p = V, 
V heing a variable quantity, then 
30. Apparently this is the most general theorem which exists uith relation to that 
class of transcendents in which a perfectly arbitrary symbol of functionahty occurs under 
the sign of integration. If we specify the form of f so as to meet the case of the 
particular transcendents discussed in the previous sections of this paper, we shall obtain 
results accordant with, but less general than, those which have been there obtained. 
But the most important feature of the theorem is, that, without restrictmg the generality 
