PROFESSOR CAYLEY’S ADDITION TO MR. ROWE’S MEMOIR. 
753 
/X] values of a function containing a radical a ^ 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 1 fx l 
coefficients Aj are different from each other:* the like remarks apply to the other 
factors. It applies in particular to the term \y—x K j , viz., it is assumed that the 
coefficients A in the X6 series y= Aah+ , . . , are all of them different from each other. 
These assumptions as to the leading coefficients really imply Abel’s assumption that 
—, . . . — are all of them fractions in their least terms, and in particular that - is a 
fraction in its least terms, viz., that X — 1: I retain however for convenience the 
general value X, 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(cc, y, z) 
of the coordinates, which on putting therein z=l becomes =x( x > V)- The equation of 
the curve thus is F(:r, y, z) = 0; and the order is = 1 ?r 1 /x 1 + . . . -|-Xd-|-« 5 /x 5 + . . . , 
/A 
=m 1 n l + • • • + Xd + J2g/X 5 -f- • • . ; viz., if K is the order of the curve x( x > V)~ 0, then 
’K='Z'nm-\-X6-\-'Z"nfi. 
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 (writing 
for convenience y = 1) is denoted by the function 
X m i j 
where observe that the expressed factor indicates n 1 branches (z—) , or say 
m x m , 
partial branches z— tc’^-^that is ^(^ — /Xj)partial branches z— Apc’^-Mi-j- ..., 
with in all n 1 (m 1 —p, : ) 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 
. . . ; 
/ Ms )» 
where observe that the expressed factor indicates n 5 branches ( z — y ) , or say 
* This assumption is virtually made by Abel, p. 198, in the expression “ alors on aura en general, 
excepte quelques cas particuliers que je me dispense de considerer: li(y’—y") = hy’, &c. viz., the mean- 
ing 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 coefficieuts of the two leading terms 
paust be unequal. 
5 fi 
MDCCCLXXXI. 
