MR. R. C. ROWE OX ABEL’S THEOREM. 
in other words, our function must not reduce itself so as to contain x only. This is 
clear a priori; for if it should so reduce itself we might choose for q Q a linear function 
of x, which is generally impossible (art. 5). 
It will be useful to examine at what point the assumption vitiates the subsequent 
demonstration. We should, in fact, have 
so that 
E = %i)% 2 ) • • • %») 
= 'Mo • ■ Wo 
=% n 
and this vanishes for all the values of x obtained by putting E = 0, so that the right- 
hand side of the equation 
f(x, y)dx= 
/ 2 cz ) f 0 o *)F(*r xXvr) 
is identically zero, and the whole process invalid.* 
* There is one case in which the function 0 may be legitimately reduced to the single term q 0 ; viz. 
the case when x is a linear function of y. 
It is plain that, as n — 1, we have not the difficulty of repeated roots which generally vitiates the result 
of this assumption. 
In fact, let 
. x(ii)=y + * 
while 
Then 
and, as by p. 718, 
we have 
6 — clq -j- a^x -j -. . . + x" 
E=F(a)=0 
cF(® 
Y\x)~ 
se 
xX 
2 ydx=^ ■ cla 0 +'2 w -da 1 + ... +2 
OCX 
1 
0' 
cla„_ j 
As an example of which formula, let x—x 
so that 
Put vi —0 and we have 
But 
whence 
while 
+ 1 
+ i — I 2 0 tC ^' o+ l“W^ ai+ ■ ■ ■ + 1“ & clan -^ 
-^■=| 4 w+l 4 wi+ • • • 
- 2.r = a„_i 
2--, = 0 if lc<n — 1 
And this is the theorem (easy to prove otherwise) which was assumed in the course of the general 
demonstration on page 719. 
