ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 343 



and from this that 



^—J^=y' . . . ; (2) 





Combining these equations together, we have 



from whence we obtain 



cHogJI. f 11=. /jl^ _^J^„.i+lUvnog.2*, 



°^ I f (d log, ^)^ • f J V I y" ■ (f^ loge qy f- r ^'^ ' 



which immediately leads to the required differential equation. A similar 

 investigation applies to y=\ / "" , y= . / ""^ . 



We may find a more general solution of Jacobi's differential equation thus 

 It Ls easily proved that if k and k' are interchanged, equation (1) is un- 

 altered, and therefore K' is a particular solution of it. Therefore 



are also solutions of equation (1), and consequent^ 



d.lr dP-\ r J M TT J 



Moreover, it is proved in the ' Fundamenta ]S"ova,' p. 74, that 



d. 



.(^'J^'a' + hhyUog,^^ 



These equations exactly correspond with (1) and (2), and therefore 



V ns/aa' + bb' ^ 



satisfy Jacobi's differential equation. 



Section 5. — In close connexion with this paper is another left by Jacobi 

 among his manuscripts, " Darstellung dor elliptischen Functionen durch Po- 

 tenzreihen," published in the fifty-fourth volume of Crelle's Journal. Its 

 object is to expand the four functions d{.v), d^(.v), d^(.v), e^(x) in terms of (x). 

 I shall give the outlines of the investigation for d.^x). 



* Because 'IJ^S.^^^ ! 1 _ l^ JP_^ ^TZr^j^^^^HlH . I have followed 



a route somewbat^differing from Jacobi's, and easier if we only wish for the differential 

 equation for^?]5h. I should advise the reader to follow the method fort/ = \/?E indi- 

 cated here, and then to read Jacobi entire. 



