ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS. 259 



a rational function of x and any one of the n roots y^ {i=l, 2, . . . n) of 

 the equation 



the p's being integral functions of x ; then X (see Liouville's paper men- 

 tioned above) may be given one or the other of the forms 



■wherey(cc) is a rational function of x, and <p,{^) ^^ ^^ integral function of 

 X and y.. 



Two leading questions are considered : (1) To find the cases in which 



one can express X.dx by a finite number of algebraic and logarithmic 

 operations (see art. 34) ; (2) To find the relations among the integrals 



\f{^i)fX^\)d^u \f{^2)l'h{^2)dx.2, . . . 



which correspond to variables Xi, x^, . , . that depend upon one another, 

 and upon the different roots y^, y^, . . . 



Broch, ' Memoire sur les fonctions de la forme,' 



ix'-y^-'fix^) {R{x'-)Y'Pdx,' 



(Crelle, bd. xxiii. p. 148, 1841), developed rules for the summation of the 

 transcendents mentioned in the title, where y (a;'') is a rational function of 



x", y an integer which is dirisible by , r and p are integers, and s an in- 



y 



teger less than r. p. 



These are analagous to the investigations of Abel on the hypei'elliptic 

 functions which had already been published. 



In a previous memoir (Crelle, bd. xx. p. 178), Broch had discussed the 

 special case where ^?=1 and s=--l. The basis of this paper is Abel's 

 memoir, ' Demonstration d'une propriete gdn^rale,' (fee. Broch also sought 

 the minimal number of integrals (Abel's y), through which a sum of inte- 

 grals could be represented. 



Minding divided his paper, ' Propositiones quaedam de integralibus 

 functionum algebraicarum,' &c. (Crelle, bd. xxiii. p., 255), into three heads. 

 He first gives an expression for a sum of integrals of the form 



{ <Po (a^i) F (a ;^, y< jpJ dXj 

 J <t>(^) 



when fg and F are integral functions, and 



(J){x) — {X — Ci){x — C<^) . . . (x — c^), 



the c's denoting constants, x^ and yi are the common intersections of two 

 curves 



i'o2/"+^i2/""'+ • . • +Pn=0; 9i2/"-'+?22/"~^+ • • • +?-.=0, 

 which correspond to Abel's ■)^(x,y)=:0 and 6{x,y)=^0. 



Minding further allowed arbitrary variable parameters in his functions 

 q, so that his results, as Brill and Nother • remark, are only less general than 

 those of Abel in that fixed points of intersection of the two curves are not 

 considered. 



' Jahrether. der devtichen Mathematiker' Vereinigung, bd. iii. p. 229. 



S2 



