ON THE HISTORICAL DEVELOPMENT OF ABELIAN FUNCTIONS, 269 



In this connection we note a paper by Richelot, * Ueber die Reduction 



des Integrales — ui=4^= auf elliptische Intesrale,' Crelle, bd. xxxii. 



p. 213. . 



In the third supplement of the ' Traite des fonctions elliptiques,' 

 p. 207, Legendre investigated certain special forms of Abelian integrals as 



and calculated the values of such integrals with fixed limits 



o' 



on p. 333 (Joe. cit.), by making use of certain substitutions, he showed 



/■•'■ 



that the integral I :^= could always be reduced to two 



I ^/.»•(l-x2)(l-/c2a;2) 



• 



elliptic integrals of the first kind that have the same amplitude, and whose 

 moduli are the complements of each other (cf. a memoir by Catalan, that 

 was crowned by the Academy of Brussels, ' Mem. couronne's par I'Acad. 

 Royale de Brux.'xiv. 2nde partie, p. 2). 



Jacobi (Crelle, bd. viii. p. 416) extended this theorem to the integrals 



, and to integrals of the more general form 



C dx , f xdx 

 — = and - , 



] V^{x) ]^/R{x) 



where E. (x) = a;(l — x) (1 — k\x) (1 + kx) (1 + Aa;). 



He showed that such integrals may be always expressed as the sum of 

 two elliptic integrals of the first kind which have the same amplitude, 

 but in general different moduli. See also Gauss, ' Determinatio attrac- 

 tionis, quam in punctum quodvis positionis datae exercet planeta,' &c. 

 ('Werke.'iii. p. 333). 



More recent e.vamples ' of similar reductions are given by Hermite 

 ('Ann. Soc. Scient. Brux.' I., B, p. 1, ' Comptes Rendus,' t. xl. 1855); 

 John C. Mallet (' Trans, of Dublin,' 1874) (he extends the theorems of 

 Jacobi to hyperelliptic integrals of any kind) ; J. C. Mallet (Crelle, 

 bd. Ixxvi. p. 79, and bd. Ixxix. p. 176) ; Cayley ('Compt. Rend.' t. Ixxxv. 

 pp. 265, 373, 426, 472). 



(30) Richelot (Crelle, bd. xii. p. 181) shows that integrals of the form 



/J x (■ 'F(x)dx 



} s/ (A i-Bx + Cx-'){Ai + B^x + Cix){A,-\- B^x + C^x^) 



' We mention in passing Gordan, ' Ueber die Invarianten biniirer Formen hiiheren 

 Transformationen ' (Crelle. bd. Isxi. p. 164); Arondhold, 'Integration irrationalcr 

 Diflferentiale' (Crelle, bd. l.xi. p. 95). In this paper extensive use is made of invariant.'!. 

 Bvioschi (Com^^t. Rend. t. Ivi. and t. lix.) bases this theory of the reduction of integrals 



I ¥(x,y)(lx upon the theory of the covariants of the ternary form. See also Brioschi 



{Compt. Bend. Ixxxv. p. 708, 1877) ; Konigsberger (Crelle, bd. Ixiv. p. 17; bd. Ixv. p. 335; 

 bd. Ixvii. p. 97 ; bd. Ixvii. p. 56 ; bd. Ixxxv. p. 273 ; bd. Ixxsix. p. 89 ; Math. Ann. 

 bd. XV. p. 174) ; Bolza (iJ/a<Zi. Ann. bd. xxviii. p. 447). A somewhat extended account of 

 the reduction of hyperelliptic integrals, including many of the more recent investiga- 

 tions, is found in Enneper's Elliptisclie Functionen, p. 501 et seq. 



