MR. R. 0. ROWE ON ABEL’S THEOREM. 
733 
We have then 
(ii.) For fi. 
Since 
it follows that 
So 
Now 
a—tq+n— 1 — F 0 . 
■ ■ ■ %.)=FoF 
%Q{y)= F 0 +F = F 0 +p, 
[x—a='Z l d(y) — %q— n+1.* 
%)=?#" 
=q,+ry 
and it becomes necessary to find y x , y 2 , . . . y n . 
14. We require the following Lemma. 
The quantities y x , y 2 , . . . y n , are, in general equal in sets. 
7)1 
For let y x =—; this being a fraction in its lowest terms (and we will take the 
denominator positive). 
Then one root of y being, when expanded in descending powers of x, 
m x 
y=Ax>h-\- . . . 
the expressions 
m x 
y=A(i) x xF-{- . . . 
771 j 
y— Aajyc' i i+ . . . 
y— &c - 
(where 1 , oj x , co 2 . . . are the /x th roots of unity) are also roots, and if these are y. 2 , y 3 . . . 
we have y x =y z = . . . , the number equated being clearly a multiple of g x . Let it 
be n x g x ; and write 
Vi-y%= 
• • • =y* x 
where 
h x — n x [x x 
y*.+i = 
• • • =yit. 
where 
h /i ^ '—’ ^2/^2 
&c. = &c. 
2^+1= • 
• • = J h 
where 
k — h~i — n/jx/ 
and 
h = n\ 
* Here 6(iy) means the degree of 6{y) when rendered a function of x by substitution for y from the 
equation x(?/)=0. 
f This lemma is the second of the theorems laid down by Abel in his important memoir “ Sur la 
resolution algebrique des equations,” of which consists the last article (it was never finished) in the 
second volume of his works. 
