772 PEOPESSOE BOOLE OIs” THE COMPAEISOX OP TEA^s’SCEXHEXTS. WITH 
wherein r is a rational function of w, and of any new variables «!, « 2 ? • • tenns of which 
the value '2^f{x,y)dx is to be found*. Juegexsex has remarked that this problem may 
be reduced to that of the determination of the value of the expression 
^y{x)<p{x,y)dx, 
f{x) being a rational function of x and ^{x^ y) a rational and integral function of x and y ; 
and under this form, adding also the restriction that the coefficient of the highest 
power of y in (1.) shall be unity, he has solved the problem. Mixdixg has investigated 
the solution when the above restriction is not imposed, but his analysis is in reahty 
founded upon a transformation in which jy^y takes the place of yf . 
We can, both with increased generality and with that gain of simplicity which results 
from the employment of the symbol 0, solve the same problem by the method of this 
section. But as the comparison of the algebraical transcendents is not the most important 
object of this paper, I do not propose to enter here upon the investigation in its most 
general form, but shall demonstrate a theorem which, while it is sufficiently general for 
all practical ends, will at the same time serve to throw light upon a peculiarity in the 
theorem of art. 22 already demonstrated. 
27. Problem.- — Mequired, infinite terms, the value of the integral expressioyi 
l,^^{x)ydx, ( 1 .) 
ip(x) denoting a rational function of x, and y an irrational function of x determined hy 
an equation of the mth degree, 
( 2 .) 
where Pi..p„ are rational and integral functions of x. We shall suppose the different 
simultaneous values of x wider the sign 2 determined hy an equation 
y=x^ ( 3 .) 
being a rational function ofx of the form -5 in which u and v are polynomials. The 
value of the integral expression (1.) is to be found in terms of the coefficients of those 
polynomials. 
The equation (3.), cleared of the radicals contained in y, and arranged with respect to 
the powers of %, will be 
V/ 
or writing for % its value -■> and clearing of fractions, 
PoU'^-\-piU"^~^v..-\-pjf''=t) (4.) 
This equation, on substituting for u and v their values as polynomials, is rational and 
integral with respect to x, and occupies the place of the equation E=0 in the general 
theorem of transformation. We shall suppose it of the wth degree. It may of course 
♦ Abei’s Works, vol. ii. p, 66. 
t Creele’s Journal, vol. xxiii. 
