1880.] Mr. R. C. Rowe. Memoir on Abel's Theorem. 517 



The notation is this : — 



We first, to escape the inconvenience of irrational and fractional 

 forms, introduce two functions, /, x? where X, the subject of integra- 

 tion =/(», y), y being determined as a function of x by the equation 



where _p l5 p 2 • • • P» are national integral functions of x, and % there- 

 fore a rational integral function of x, y ; while / is rational, but not 

 necessarily integral. This assumption will cover all cases. 



1 



[For example, in the case X=— y=====i 



v± — X 2 



we have the equations f(x, y)=\ 



y 



xfe2/)=2/ 2 -(l-^)=0]. 



We next change f(x, y) so that its denominator shall be the pro- 

 duct of x(y)> the differential coefficient of % with respect to y, and a 

 function of x only, and write 



/ 2 0) -.x'G/)' 



This is shown to be always possible. 



We then introduce the equation whose roots are the variables of the. 

 functions we add. This equation is obviously not arbitrary. It is 

 necessary and sufficient to take the result of eliminating y between x 

 and any other function of x, y which by the use of % can be made of 

 (at most) the n— 1th degree in y. 



Let this function be 



%)=2«-i2/ tt ~ 1 + • • • +ffiy + 2o» 

 and let the result of elimination be called E. 



Here q n _ v q n _% . . . g lt q are rational integral functions of x, and E 

 is a function of x and of their coefficients. 



It may happen that, owing to relations between these coefficients, 

 the equation E = is satisfied by values of x independent of them. 

 Let F (aj) be the product of the corresponding factors, and write 



■E(x)=¥ (x)¥(x). 



It only remains to define a symbol 0, used by Boole in his paper on 

 the comparison of transcendants. The following is his definition : — 

 " If («)/(«) be any function of x composed of two factors 0(#), 

 /(*), whereof 0(V) is rational, let 



e[<A0*)]/O) 



denote the result obtained by successively developing the function in 



2 2 



