31 
713] 
memoir on abel’s theorem. 
groups, each a product of fij series; and in each such product the coefficients A, 
l 
are in general the values of a function containing a radical and are thus 
different from each other: it is in what follows in effect assumed not only that this 
is so, but that all the n^fi x coefficients A x are different from each other* : the like 
remarks apply to the other factors. It applies in particular to the term [y - x K 
a 
viz. it is assumed that the coefficients A in the \6 series y — Ax K -\-... are all of 
them different from each other. These assumptions as to the leading coefficients 
• • Tit Tit-fa 
really imply Abel’s assumption that — 1 , ..., —' are all of them fractions in their least 
/L yk 
terms, and in particular that - is a fraction in its least terms, viz. that \ = 1: I 
A 
retain however for convenience the general value putting it ultimately = 1. 
In the product of the several infinite series, the terms containing negative powers 
all disappear of themselves; and the product is a rational and integral function 
F(x, y, z) of the coordinates, which on putting therein z=l becomes = ^(x, y). 
The equation of the curve thus is F (x, y, z) = 0; and the order is 
viz. if K is the order of the curve % (x, y) = 0, then K = Snm + \0 + X"n/x. 
The curve has singularities (singular points) at infinity, that is, on the line z — 0: 
viz. 
First, a singularity at (z — 0, x = 0), where the tangent is x — 0, and which, 
wilting for convenience y = 1, is denoted by the function 
m Y 
say n x (m x — yi) partial branches z — x m ' ^, that is, n x (m y — /¿j) partial branches 
£ = A x x m ^~^ + ..., with in all n x (in x — ¡x x ) distinct values of A,: and the like as regards 
the unexpressed factors with the suffixes 2 and 3. 
Secondly, a singularity at {z— 0, y = 0), where the tangent is y = 0, and which, 
writing for convenience x = 1, is denoted by the function 
* This assumption is virtually made by Abel, (/. c.) p. 162, in the expression “alors on aura en général, 
excepté quelques cas particuliers que je me dispense de considérer: h(y'-y")=hy r , &c.”: viz. the meaning is 
that the degree of y' being greater than or equal to that of y", then the degree of y'-y" is equal to that 
of y" : of course when the degrees are equal, this implies that the coefficients of the two leading terms must 
be unequal.