conwell: a special reimann surface. 47 



values a^o and a.%, that as a., and a 3 approach a'^o and a^^ ^^ 

 value, one or more holes in the surface must be continually- 

 decreasing in size in such a way that when a", and a^^ are 

 reached the surface has a node at the point (i\„ y,„ n^) on F 

 and vice versa. If {x^, ij^, u^, i\) is the corresponding point 

 on $, the latter will also have a node at this point. Therefore 

 corresponding to nodes on F are nodes on $. At such nodes 

 the tangent hj^jer-planes to 



P {x, y, u, I') = 

 and 



Q {x, y, u, r) = 



are coincident. In order to investigate the nature of F at such 

 places write the equations of the tangent hyper-plane^ to P 

 and Q at the point {x^, y^„ z^^,, vj, and the conditions fo.- their 

 coincidence. The equations in question are, 

 {x-Xo)P'x^-\-{y-2jo)P'yo-^{v-Vo)P'eo-r{u-u„)P'u, = 0, (23) 

 and 



(x-Xo)Q'xo + (?/-?/o)Q'yo + (M-Mo)Q'uo + («'-?'„)Q'vo = .(24) 

 The conditions for these two hyper-planes to be coincident is 

 that 



Q'xo QVo Q'uo ~ Q'vo 

 It is e\ident, however, from the relation 



P {x,y,u,v) + iQ{x,y,u,v) = 

 that 



P'xo = QVo, PVo= -Q'xo. P'u., = Q'vo, and P'vo= -Q'uo. 

 Hence 



P-'xo+Q-'x,, = 0, P-'yo+Q"y.,0, = P-'u„+Q-'u, = and P-'vo+Q-'eo = 

 and therefore 



Px'o = PV. = P'u. = Pvo = Q'xo = Q'yo = Q'uo = Q'vo = 0. 

 In the above relations 



P = u- — V- — S {x, y) 



and 



Q = 2ui' — t (x,y), 



therefore it follows that u = and r ^ and therefore that 

 g (z) =0, Moreover, since 



P'xo+?Q'xo = and P'yo+?Q'yo = 



