ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 315 



Etiuating coefficients in the two expressions for 0^ &e. in (1) and (5), we 

 have 



cUr 1 -f .. I .... (C) 



where the coefficient 



^. ,. ^. f,^ /-, IN in(m — 1)...()H — « + l) /»-N 



m, = (2i + l)i2t + 3)...(2m-l). ^ i.2.3...z: ' * ' ^'^ 



■we find from (G), in consequence of (3), that 



{(2»i+2/ + l)iH;-(m + lMr;+(m-i+2)»ii_2?Ji',-2 



db da 



which being transformed by means of (7), gives 



.,-(,--iXo;_3)i,-«.i{»%i-i6%:}, 



which is independent of (m), as it should be, according to the assumption. 

 Moreover the value of „,„, , found by equating coefficients in (1) and 



(5) is 



CZ-+VA ^ A^^r.,„ , 1^■R-+l^A„^1^ ..-n'—A^ 



dli 



m+l 



A-^{(,n + l)^B'»+' + (m + lX/-,B'"*'A='+ . ..}; 



and this can also be found by difierentiating (6) with regard to'^. The induc- 

 tion is completely proved. 1 think the reader will find no difficulty after 

 these remarks in reading this paper, — one of the most beautiful productions 

 of its illustrious author, the paper reviewed in last section having been 

 previously studied. 



Section 6. — In the thirty-seventh volume of Crelle's Journal there is a me- 

 moir of Jacobi of a very different nature to those we have been reviewing. 

 Its title is " Ueber unendliche Eeihcn, deren cxponenten zuglcieh in zwei 

 verschiedenen quadratischen Formen enthalten sind." Many parts of this 

 memoir relate to the theory of numbers, and contain no allusion whatever to 

 the subject of this lleport ; nevertheless, as these investigations were sug- 

 gested by a formula in the ' Fuudamcnta Nova,' it will be right to give some 

 account of it here. 



From the well-known formula in the ' Fimdamcnta Xuva,' 



(l_5=)(l_2'Xl-r/). . . (l-5r)(l-./--Xl-r/,^). .. 



(l-^.-')(l-2^r-')(l-r/.-'). . .= 



l-qiz + z-^) + q\z"- + z-^)-q\z' + =-^)+ . .., 



we deduce, by putting 2" for q and ±q^'' for z, the following expressions: — 



n{(i-r/''"+'"-")(i— 2^"''+'"+''Xi— 5-'"'+-''')}=2:(--i)V"''+"''.. . . (I.) 



n{(l + 22'"i+'"-'')(l + y2mi+m+H^(]^_^2m.+2m^^ =Vj'»>=+"'. . (II.) 



It is easy to see that these equations will lead to a multitude of others of 

 1869. 2 A 



