of Edinburgh, Session 1874 - 75 . 
441 
was originally designed, this result is very curious, as suggesting a 
form of the square root of the operation of simple integration. In 
fact it gives 
( )=4 rui j 
o % J x ~y 
Seeking to obtain an elementary proof of Abel’s result, which 
should at the same time be applicable to any function, whether 
developable or not, I hit upon the simple expedient of inverting 
the order of the two integrations. We thus get the proof im- 
mediately in the form 
dy/XQdi = r‘ r‘ 
Jx-yJy-t A A Jx-yJy-t 
Now it is known (and a simple geometrical proof is easily given) 
that 
dy ' 
x-y Jy-£ 
7 r . 
Hence the integral becomes at once 
»[/(«) -/(°)] • 
Numerous extensions and applications of the theorem are given. 
As one example of these extensions the following, which assigns 
an expression for 
in 
, may here be given- 
r* aa doc ^ p x 2 dx 3 
Jo (x 2 - X.f^rT 
■s. 
•«. f(.m 
n — 
(Xn - 
= E 1 E 2 E,_, [/(*,)-/((>)]. 
