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VIII. On Abel’s Theorem and Abelian Functions. 
By A. Id. Forsyth, B.A., Fellow of Trinity College, Cambridge, 
Professor of Mathematics in University College, Liverpool. 
Communicated by Professor Cayley, F.R.S. 
Received October 28,—Read November 23, 1882. 
The present paper is divided into two sections. The object of Section I. is to obtain 
an expression for an integral more general than, but intimately connected with, that 
occurring in Abel’s theorem. The latter, as enunciated by Professor Pv,owe in his 
memoir in the Phil. Trans., 1881, is as follows:—If 
x( x > y)=° 
be a rational algebraical equation between x and y, then an expression can always be 
found for 
where f(x) is a function of x only, U a rational algebraical integral function of x and y, 
and the upper limits of the series of integrals are the roots of the eliminant with 
regard to y of x{x, y) = 0 and a function 0(x, y). 
In the case here considered two equations respectively of the degrees m and n 
between three variables 
F m (x, y, z) = 0 
F n (x, y, z) — 0 
are given (these alone being considered, as subsequent generalisation to the case of r 
equations between r dependent variables and one independent variable is obvious); 
and an expression is obtained for 
l 
Udx 
p 
- 1 - Mh 
F„ 
y, * 
the upper limits of the integrals being given by the roots of the equation arrived at 
by eliminating y and z between F w , F„ and an arbitrary equation 
Ffx, y,z )=0 
or, what is the same thing, by the co-ordinates x of the points of intersection of the 
three surfaces represented by F m , F„, F ;; . 
2 t 2 
