734 
MR, R, 0. ROWE ON ABEL’S THEOREM. 
Also let us write, for shortness, 
vu 
Too 
— °"lJ — - 2 > 
Pi Pi 
Pi 
=or, 
and let these be in descending order of magnitude, so that 
cr, > Co > cr, . . . > 07 . 
Tft 
We have then n x sub-sets, each of /y terms, with index —, n z sub-sets each of 
Pi 
m,o 
terms, with index —, and so on. 
Pi 
These quantities m x , /x,; m 2 , /x 3 ; &c., can be speedily determined when y is given by 
r Newton’s method. 
Thus, write A.W for y in the equation, and determine cr by the condition that in the 
resulting function of x the indices in two or more terms may be equal and greater 
than in any other term (while the condition that the sum of these terms shall vanish 
will determine A).* 
These conditions are obviously necessary for the existence of a root y=Ax a -\- . . . : 
and it is easy to prove directly that we can thus determine values of the quantities <x 
unique, and in descending order. 
For suppose the indices after substitution to be ncr\ (n — l)<x+cq; {n —2)cr+a 3 ; . . . 
Then putting 
ncr— (n — k :)<x+cq. 
we have 
cr= 
Ok 
k ’ 
* As an examjile, suppose that y is determined by the cubic 
x=y z +PiV 2 +ihv +Po= 0 
r i=3; Bo= 2 - 
3cr, 2<7-1-1, <7 + 3, 2. 
while 
Writing Kx* for y the exponents are 
It is clear that the conditions are satisfied by making 3<7=<7 + 3, i.e., <7=f, while a quadratic is obtained 
for A, so that there are two corresponding’ terms and Vi-Hy 
They are also satisfied by making <7+3=2, i.e., <7= —1, and a simple equation is obtained for A. 
We have, then, 
—°i—f- j n \ —1, 
J“i 
— 2 = <7 __ 1 . n _ 1 
Pi 2 
