MR. R. C. ROWE ON ABEL’S THEOREM. 
739 
Writing down a series of equations 
similar to (C) we have 
%i) 
= [pi]+di cr i = [« — !] 
+ (« 1)CT y 
+ A m _ x 
- 
-N 
=M+/ > i°T=[ n — 2 ] 
+ (n — 2)cr 1 
+A n _ 2 
First set 
&c. 
= &c. = &c. 
+ (n— kjcr-y 
+ A n _; Ci 
- 
%**+1) 
= [/b]+ PF ' 2 = [n —k y — 
l]+(w —Aq— 1) 
o' 2+A n _/ fi . 
-1 
Second set 
&c. 
= &c. =&c. 
- 
&c. 
II 
O 
II 
O 
]&c. 
and, adding all these lines together, 
^{y ) = [_ n ~■ l]+[ w —2] + . . . +[0]+(w—1 + . . .-\-n—k 1 )a 1 
~h(n— /q—1+. . . -|-n— k 2 )or, 2 -\-. . .+ SA. 
or 
%0y—%q={n— 1+. . .-\-n—k 1 )cr 1 -\-(n—k 1 — 1+. . — & 2 )cr 3 +. . . + SA 
Now, if the condition of (i) were not satisfied, some at least of the signs of equality 
connecting the first and second vertical columns must have been replaced by the 
sign >; and as those between the second and third column would have remained as 
before, the equality at the head of this page would have become an inequality— i.e,, 
the value of XOy—Xq would have been greater than it is— i.e., y — «. would have been 
greater. 
It only remains to consider the term 2 A. 
The smaller we can make this sum, and therefore, all the terms being positive, the 
smaller we can make each term, the less will be our value of y — a. 
Now from the general equation 
M + /R°V= [«] + acr A + A a 
we see that, since [p A ] and [a] are integers, A a consists in general of two parts—an 
integer and the proper fraction which added to (a — p A )cr, v will make it integral. 
Now we can make the integral part vanish for every value of a ; for to do so will 
only require a relation between the major term and the other terms of each set; 
so that, given the degree in x of the major term, those of the others in its set can be 
written down. 
As the conditions (i) only connect with one another the major terms of different 
sets, this last condition is independent of them and can always be satisfied. 
5 c 2 
