588 



Prof. A. R. Forsyth. 



[Nov. 23, 



-when the magnetising force is so high, as to give the wire the maxi- 

 mum magnetisation ; thus confirming beyond all doubt what has been 

 pointed out theoretically by Thomson (" Electrostatics and Magne- 

 tism," § G67) and indicated experimentally by Rowland. 



III. " On Abel's Theorem and Abelian Functions." By A. R. 

 Forsyth. B.A., Fellow of Trinity College, Cambridge, Pro- 

 fessor of Mathematics in University College, Liverpool. 

 Communicated by Professor Cayley, F.R.S. Received 

 October 28, 1882. 



(Abstract.) 



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 Mr. Rowe in his memoir in the Phil. 

 Trans., 1881, is as follows : — If 



x(*, y)=o 



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 is a rational algebraical integral 

 function of x and y, and the upper limits of the series of integrals are 

 the roots of the eliininant with regard to y of x(#> y) = Q an ^- a func- 

 tion 6(x, y). 



In the case here considered two equations of the degrees m and 

 n respectively between three variables 



¥ m (x 9 y,z)=0 

 F n (x, y, «)=0 



.are given (these alone being treated, as subsequent generalization to 

 the case of r equations between r— 1 dependent variables and one 

 independent is obvious) ; and an expression is obtained for 



v r ™- 



the upper limits of the integrals being given by the roots of the 

 equation arrived at by the elimination of x and y between F Wi , F« 

 ..and an arbitrary equation 



F p (x, y,z)=0, 



