MR. R. C. ROWE ON ABEL’S THEOREM. 
hi r 
/ 1 o 
Section I. 
1. The general question to which an answer is sought in what is called the Theory 
of the Comparison of Transcendants may be stated thus:— 
Is it always possible to establish, between the values for different variables of the 
integral of am algebraic function hoivever complex, algebraic relations: the variables 
themselves being connected by any requisite algebraic laws P 
If, for example, 
jXcfe=F(x) 
where X is any algebraic function of x, rational or irrational, integral or fractional, is 
it necessarily possible by connecting x x , x 2 , . . . x n by any requisite algebraic laws to 
obtain an algebraic (or logarithmic) expression for the sum 
F(* 1 )+F(x 2 )+ . . . +F(x„) ? 
This question is suggested on the one hand by such well-known results as 
and 
if 
Ffoj + F (x 2 ) = constant, where X=—/p == p', if xp-\-xf=l 
F(xj) + F(x c ) + F(x 3 ) = 0 where X== 
ffl-xil -k 3 X* 
4( 1 — xf) ( 1 — xf) ( 1 — xf) = (2 — xf — x 2 z — xf + Jdxpxlx 3 2 ) 3 , 
and on the other hand by the possibility of finding algebraical expressions for many 
symmetric functions of the roots of equations though these roots may not separately 
be so expressible. 
It is in fact this combination of the theory of integrals and the theory of equations 
which furnishes the key to the problem ; enabling us to express the requisite algebraical 
laws very concisely by a single equation of which the variables are roots, and whose 
coefficients are not independent but connected by a corresponding number of relations. 
2. The expression of the function to be integrated. 
To escape the inconvenience of fractional and irrational forms we first introduce two 
new functions and a fresh variable. 
Whatever be the nature of the function X—the subject of integration in the 
transcendants we are considering—it may be written 
A x > y) 
4 z 2 
