ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 261 



function in ascending powers of each distinct simple factor (of the form) 

 x—a in the denominator of </'(■<■), taking in each development the coefficient 



of , adding together the coefficients thus obtained from the several 



x — a 



developments, and subtracting from the result the coefficient of l/x in the 



development of the same function in descending powers of .x.' 



The simplifications made by Boole are more than offset by the loss of 

 generality which characterise his formulae. 



Rowe (Memoir on Abel's Theorem, 'Phil. Trans.' 1881, p. 713) 

 endeavoured to simplify Abel's results, and at the same time to retain 

 their generality. Using the same notation as in art. 18, and employing 

 Boole's symbol, he derives Abel's theorem in the form : 



=0[. 



I 1 F,, (x) "sM^yl log B(y) + C, 



■where C is a constant. 



Professor A. R. Forsyth, in a paper, 'Abel's Theorem and Abelian 

 Functions' ('Phil. Trans.' 1883, p. 323) has obtained an expression for an 

 integral that is more general than that occurring in Abel's theorem. 



Professor Forsyth takes tM'o given equations of degrees m and n 

 between three variables, of which y and z are dependent, x being 

 independent : 



F„.(.T, y, c) = 0, F„(,r, y, z)=Q. 



The Jacobian (functional-determinant) of these two functions is 



/F F \ 

 denoted by 3 i—^!^ — -\. A quantity T is defined by the relation 

 \ y, ~ / 



rj,_ ^(.x,y, z) 



*(•«, y, 2)' 



where ^' and * are rational algebraic functions of x, y, z. This quotient 



may in turn be expressed in the form — rr— , , where U is an integral 



function of x, y, and z, and./(.x') a function of x alone. 



The generalised Abel's theorem, as derived by Professor Forsyth, is 



Vdx ^V I 1^{ U 



+ C, 



^J/(.)j]£f) ®[/(-)]^- 



F„„ Fh 



loffF 



o ■*■ i' 



y,^ J \ \ y,- 



where the upper limits of the integrals on the left-hand side are the m.n.p 

 roots of the equation obtained by eliminating y and z between F,„ and F„, 

 and an arbitrary rational algebraic function Fj,(a:,i/,s)=0. On the right- 

 hand side tlie summation extends over the m . ti roots y and z in terms of 

 X of the equations F,„ = and F„=^0. C is a constant. 



The more general theorem Professor Forsyth enunciates as follows : 

 Let F,(a;|, x^, . . . x^) = {i=l, 2, . . . r — 1) be r— 1 algebraic equations 

 of degrees m^, m^, . . . m^_j^ respectively, giving .Xj, x-^, . . . x^ in terms of 

 a, ; and let F,.(a;2, x^, . . . x,) be a function of these dependent variables, 

 the coefficients of which are functions of x, containing any number of 

 arbitrary constants. Form the eliminant E of all the F's, so that we 



