MR. R. C. ROWE ON ABEL’S THEOREM. 
717 
So fix, y) the subject of integration is reduced to the form 
/iCl ?/) 
mxiy) 
in which it will hereafter be used. 
5. The equation whose roots are the variables of the functions we compare. 
This equation is clearly not arbitrary ; for if it were we could choose it linear ; and 
having then only a single integral, should be required to find for it an algebraic (or 
logarithmic) expression, a thing generally impossible. 
We shall find it sufficient to take, for this equation, the result of eliminating y 
between y and any other integral function of x, y; which, by the use of y, can, of 
course, be made of (at most) the (a— l)* 1 ' degree in y. 
Let this second function be 
1 + T-hy n 3 + • • • + 2i2/+<Zo 
and let the result of elimination, viz.:— 
%i)% 3 ) • • • %») 
be denoted by E. 
0=0 may be called the equation of condition. 
We assume q 0 , q } . . . q n _ x to be rational integral functions of x ; while any number 
of the coefficients in these functions are arbitrary : call them a x , cq, . . . 
E will then be a rational integral function of x and these quantities cq, a. 2 , . . . 
We may then either (l) take the roots of the equation E=0,—cq, cq, . . . being con¬ 
sidered absolute constants—as the upper limits of our integrals (of which alone we view 
these integrals as functions); or (2) since by a due alteration of the c/’s we may produce 
any possible simultaneous alteration of the x’s, we may consider the variables x in 
the different integrals as, in the passage from the lower to the higher limit, always 
connected by the equation E = 0, in which now cq, cq, . . . are a system of variables 
with which the variation of x has to be connected. The latter, as the more general 
and powerful hypothesis, is to be preferred. 
E = 0 may be called the equation of the limits, or the equation of transformation. 
6. It may happen that, owing to a relation connecting the as, the equation E=0 is 
satisfied by values of x independent of these new variables. This relation, since one 
of the &s of which E is the product will vanish for this value of x and 0 is linear in 
the cc’s, must be a linear relation. We v/ill then suppose 
E(cc, cq, cq, . . . ) = F 0 (£c)F(x) 
.where F 0 (,r) is independent of the «’s; and, the degree of F (x) being y, let its roots be 
