48 THE UNIVERSITY SCIENCE BULLETIN. 



it follows that 



s'xo + it'xo = and sVo + *^Vo = 0. 

 Therefore g' (Zq) ^=0, showing that z is a double root of 

 g (z) = 0. It is evident therefore that the surfaces P and Q, 

 and hence F, may be deformed in any way we please without 

 affecting its analysis situs properties provided that during 

 this deformation g (z) =0 never acquires any double roots. 

 These conditions allow a deformation that will change com- 

 plex roots into real and unequal roots without any two roots 

 becoming equal in the process. Hence we may in this manner 

 transform g (z) into / (z) , where the roots of j {z) are real 

 and unequal. 



The above conclusions show that no generality is lost in con- 

 sidering the simpler case and thereby avoiding the difficult 

 task of dealing with imaginary coefficients. The difficulty 

 introduced by imaginary coefficients is that due to the lack of 

 symmetry with respect to the XU plane. 



It is evident now that the surface constructed from the 

 simplest possible relation is sufficient for a complete exposi- 

 tion of the Riemann surface properties of the most general 

 elliptic function. 



A NUMERICAL EXAMPLE. OF THE HYPER-ELLIPTIC CASE. 



As an introduction to the general hyper-elliptic function we 

 will consider briefly a simple numerical example of the same. 

 The details of the surface F will be considered sufficiently to 

 show that the preceding discussion can be applied in all its 

 essential details to the higher form. For this purpose con- 

 sider the equation 



^o^^= iz — 5) (z — 1) (z + 1) (^ + 2) (z + 3). 

 The surface Fix, y, u)^ will be represented by 

 4^4 _ 4^25 _ yo ^ 0, 



where 



S=x^—10xHj- + 5xy''—20x^ + 60xy~—S0x- + 30ij~ + 19a: + 30 

 and 



T = 5xMj — 10a:-?/ + 2/^ — 60x-y + 20/ — 60xy + 19y. 



The surface F is symmetric to the XU and XY planes. The 

 trace on the XU plane is the XX axis and the real curve 

 v^= (x — 5) (X — l){x + 1) (x + 2) (a; + 3) 



