MR. R. C. ROWE ON ABEL’S THEOREM. 
721 
This is the general theorem for the summation of integrals of any form of which we 
were led to suspect the existence. 
It corresponds to that Numbered (37) on page 193 in Abel’s Memoir (and which 
should be called “ Abel’s Theorem,” though that name is frequently given to the very 
narrow case of it discussed on page 255), while it is more concise through the intro¬ 
duction of the symbol ©, and more intelligible through the absence of the letter vf' 
9. In general, as has been said, the function E has no factor independent of the a’s, 
i.e., F 0 (a;) = l. 
In this case the formula of the last article takes the simpler form 
fix, y)dx —® 
_/s (®). 
fed 
* xiv) 
log 
%+c 
As an example of the expansion of © suppose f 2 (oc) = (x — a) m . 
We have then 
coefficient of — in the expansion of log: 6y 
X — a L (x—a) m f \y) ® J 
u ’ o{ 
i.e., of ( _f,;' r (' z ) + ( 3 '-«i r U)+. • .} 
to 
__ i ( a ) 
i.e., = 
1 d 
TO —1 da 
6A 
%(A) g 
where A is the value of y corresponding to x= a ; and—representing by C x \(x) the 
1 . 
coefficient of - in the descending expansion of X(oc)- 
f(x, y)dx=t 
ff x >y) 
dx 
1 
d 
I TO — 1 da\ 
m —1 
Jf*, A) 
%(«) 
log 6A. I — C , 
fM y) 
—- log 6y l ~h C, 
which is identical with Abel’s formula (44). 
10. Before proceeding to examples of the use of the general theorem one or two 
points in the proof and the result should be alluded to. 
(i.) A limitation to the form of the function 6. 
In choosing this function we may not make q x = 0, q 3 =0, . . . y„_, = 0 simultaneously: 
* The want of clearness spoken of is due to an ambiguity in the important sentence (p. 187) in which 
Abel implicitly defines the letter v which is to appear prominently in his enunciation of the final theorem. 
But it is hardly necessary to dwell on a difficulty which the method of the text avoids. 
MDCCCLXXXI. 5 A 
