Dr Baker, On the differential equations, etc. 219 



On the differential equations of the hyperelliptic Junctions. By 

 Dr H. F. Baker, St John's College. 



[Received 29 July 1903.] 



The following note relates to some differential equations 

 originally published iu these Proceedings in the Autumn of 1898 

 (Vol. ix. Part ix. p. 513), of which a proof has recently been 

 published in t. xxvir. of the Acta Mathematica. It deals with three 

 points, (I) a group of differential operators which the differential 

 equations allow, corresponding to the linear group by which 

 the variable of the hyperelliptic integral may be transformed, 

 (II) for p = 1 or p = 2, a set of modular relations which is de- 

 ducible from the differential equations, which conversely would 

 furnish a proof of the differential equations, (III) with the direct 

 algebraical integration in the manner explained in the previous 

 note in these Proceedings of the differential equations for the 

 case p = l, and the degenerate cases for p = 2, p = 3 ; the general 

 case p = 2 was outlined in the previous note. The degenerate 

 forms obtained correspond with those used for two variables by 

 M. Painleve in his recent paper in the Acta Mathematica, t. xxvu., 

 on functions possessing an algebraic addition theorem ; the present 

 point of view emphasizes the existence of and gives the algebraic 

 definition of the integral function from which the meromorphic 

 functions are derivable. Comparing M. Painleve's results in 

 regard to the new transcendents found by him satisfying ordinary 

 differential equations of the second order (Acta Math, xxv.) one 

 is tempted to ponder on the existence of a theory including 

 these transcendents and the hyperelliptic functions of one or two 

 variables, to be found by a generalisation, perhaps only slight, in 

 the forms of the differential equations satisfied by the latter. 

 Such a generalisation, if not found by accident, requires a method 

 of integrating the hyperelliptic equations founded on the general 

 properties of functions, wholly different from the computative 

 method followed here. 



But such computations would appear to be often useful, for 

 several reasons, and experience suggests that they are indispens- 

 able while the general theory is still incomplete. Nor probably 

 need an apology be offered to English readers for adding as an 



