254 REPORT— 1897. 



where y" (a;) is a rational function of x and X a function of greater degi-ee 

 than the fourth in x, the name ultra-elliptic ; when X is of the fifth or 

 sixth degree in x, it is said to be of the first order ; ^ when X is of the 

 seventh or eight degree in x, of the second order, etc. In each order three 

 different kinds of integrals are to be distinguished, which are entirely 

 analogous to those ' which the nature of things has introduced into the 

 theory of elliptic functions.' 



Jacobi (' Werke,' bd. i. 1832, p. 376) wished to call these the Abelian 

 transcendents, on account of the following works of Abel that had at that 

 time appeared : — 



' Remarques sur quelques propri^t^s g^nerales d'une certaine sorte de 

 fonctions transcendantes ' (Crelle, bd. iii. 1828, p. 313). 



' Demonstration d'une propriety gen^rale d'une certaine classe de fonc- 

 tions transcendantes ' (Crelle, bd. iv. p. 200). 



These papers treat of the more special functions in which y is con- 

 nected with X by the relation 2/^=X, where X has the same meaning as 

 above ; the term hyperelliptic is usually applied to such functions, Abelian 

 being used in general to designate transcendents in which y is defined as 

 any function of x through the algebraic equation x(^> y)=0' 



(13) Abel's theorem. — A brief account of some of the fundamental 

 statements of the preceding article is given here. The mode of procedure 

 is nearer that of Riemann, Fuchs, and later writers than that of the 

 original memoir. Some of the results as derived by Abel are given later. 



The algebraic equation 



x(^>y)=po+Piy+P2y^+ • . • +;'n-ij'''"'+2/"=x(y)=0, 



in which all the coefficients p are rational integral functions of x, and the 

 integral function 



%.2/)=9o+5'iy+?22/'+ • • • +5'n-i2/"-'=%)=0, 

 where the ^''s are likewise integral functions of x, when considered geo- 

 metrically, represent two curves, which intersect in a certain number of 

 points. In the coefficients of these two equations may appear quantities, 

 -?/, V, w, . . . , quantities quite indeterminate, upon which the coefficients 

 depend. 



The co-ordinates of the intersection of the two curves are functions of 

 ti, V, w, . . , , so that we may write the two curves in the form 



X{x,y ; u,v,w, . . .)=0, 

 6{x,y ; u,v,w, . . .)=0. 



Let the points of intersection of the two curves be 



^1.2/1 ; ^2,y2 ; ^zyy-i ; • • '^^,y^ ; 



and when particular values iiP,v^,w^, . . . are given to t(,,v,w, . . . , let 

 the corresponding points of intersection be 



a^i^yi" ; a:20,y,'> ; x^^,y^<^ . . . xj^,yj>. 

 ' Legendre uses the word class. We may remark here that, when he divides in- 



tegrals of the form -—— into the three different lands, he must first assume that 

 J Vf (a?) 



n is less than- -1 or ~ , where X is the degree of <p(^x). See Richelot (Crelle, bd. 



xii. p. 185), where different forms of the integrals corresponding to the different kinds 

 axe considered. 



