744 
MR. R, C. ROWE OX ABEL’S THEOREM. 
23. We may conveniently investigate at this point, as a corollary to previous work, 
the conditions necessary that the ‘algebraic and logarithmic function’ often referred to 
already should become a constant; in other words, that the term involving © in the 
expression of Abel’s theorem should disappear, and with it the arbitrary quantities 
«i, ct 2 . . . 
We will assume F 0 (a;) = l for the sake of simplicity, and have therefore the formula 
of Art. 9. 
The first condition is that 
M x ) =1 .(A) 
for otherwise the terms contributed by it to © will introduce the arbitrary 
quantities a. 
Next, we must have 
Q^^log%=0 
xiy) & J 
or, which comes to the same effect, 
ct AMMz =0 
■ %( y) Qy 
and since 8 6y=6y, 8 indicating differentiation with respect to ci’ s, and consequently 
not altering the degree of a function in x, 
and the condition to be satisfied is 
' A ( x v) 
x(y) 
<-i 
when, for y, any whatever of the series y x , y. 2 . . . y n has been substituted. 
Now J\(x ly), being integral and rational, can be expanded in the form 
r=n— 1 
% 
r= 0 
X P ,f. 
r— 0 
We require then that, for all values of k and r from 0, to n — 1 
¥ r +ry k —fi{y,)<—\. 
Now 
xfa)=(yi-yi)&-y-2) ■ • • (^-^--i)(y*-2/* + i) • • • {yk-y n ) 
whence • • • +yi-i+(n-k)y l 
Pr<—1 +2/i+ • • • +2At--i“K n ~ k—r)y k 
so that 
