270 REPORT — 1897. 



where F(a;) denotes a rational function in x, and where the factors in the 

 denominator are resolvable into real linear factors, may be reduced by 



means of twelve substitutions of the form x= ~ into an aggregate 



c + dz^ 



of integrals 



Jon/(1 -i2^)(l -.■V)(1 - WXl^^i^^y 



where <(> (z^) is a rational function in z^, z^ being in value situated between 

 and 1, as are also the quantities k"^, X'^ and fx^. 



Richelot further proved that, whatever the degree of the function 

 under the square-root sign, provided it consist of linear real factors, integrals 

 corresponding to (I.) may be reduced to a form similar to that just 

 written.' 



He further divides these integrals into the three principal kinds, and 

 by means of Abel's theorem considers the peculiar properties of the 

 respective kinds. By making use of irrational substitutions which depend 

 upon a quadratic equation, he finds that integrals of the form 



f (M-fNs^)c^z 



J ^/ (1 -~-')(l -kV)(1 - W)(1 -^v) 



may be reduced to the same form 



f (M,-fN,2/2)dy 



J^/(l-^/2)(l-K■'V)(l-^''2')(l-A'"«') 



where the moduli are either greater or less than the old moduli. By 

 repetition of this procedure the moduli rapidly approach zero or unity 

 (cf. art. 8). 



(31) In a later paper Bichelot (Ci-elle, bd. xvi. p. 221) reduces integrals 



2dz 

 of the form C-J^ to h{^ / '" 3- ^7 



Jo Vl+x^ -Jo^(l-;:,(l-cos^;^Q.^)(l-cos2e5^,^ 



means of the transformations a;=- where t=^s/l—z^' 



Making use of a substitution that was suggested by Jacobi, 

 2i/=-Ja + bz + cz^+ Va — bz + cz^, 

 he shows that the sum and the difl'erence of the integrals 



f dx 1 dx 



}-y{a + bx + cx^){x^ + d){x^ + e) s/(a-bx+ cx'^){x'^ + d){x^ + e) 



may be expressed through one Abelian integral. The second part of this 

 memoir is devoted to the numerical calculation of hyperelliptic integrals 

 of the first order. 



In the posthumous writings of Jacobi a method is given whereby the 



' Cf. Jacobi, Getam. WerTte, bd. ii. p. 38. These integrals may be expanded in 

 converging series according to sines or cosines of multiples of the same angle. 



