conwell: a special reimann surface. 47 



values a^2 and a^^, that as a., and a,^ approach a^^ and a^^ in 

 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 (x\„ t/,^, u^) on F 

 and vice versa. If (.i\„ y,„ u^, %\) is the corresponjiing point 

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

 corresponding to nodes on F are nodes on <l>. At such nodes 

 the tangent hyper-planes to 



P {x, y, u, v) = 

 and 



Q {x, y, u,v)=0 



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

 places write the equations of the tangent hyper-planes to P 

 and Q at the point (x^„ y^, u^, v^), and the conditions for their 

 coincidence. The equations in question are, 



{x-x^)P'x, + {y-y^) P'y,, + {v-v^)P'e,, + { u - m, ) P'u,, = 0, ( 23 ) 

 and 



(a:-a:o)Q'xo + (?/-?/o)QVo + (w-Wo)Q'uo + (i'-r„)Q'v„ = 0..(24) 

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

 that 



PX _ Z^ _ Z^ _ Zx? 



Q'xo - QVo ~ Q'uo - Q'vo" • 

 It is evident, however, from the relation 



P {x,y,u,v.) -\-iQ(x,y,u,v) = 

 that 



P'xo = Q'yo, PVo= -Q'xo, P'u„ = QVo, and P'v„= -Q'u.,. 

 Hence 



P'^'xo+Q'^'x„ = 0, P^Vo+Q^VnO, = P- V.+Q-'uo = and P'^'vo+Q'''e„ = 

 and therefore 



Px'o = PV„ = P'u„ = Pv„ = Q'x.. = Q'yo = Q'u,. = Q'v„ = . 

 In the above relations 



P ^u- — V- — s {x, y) 



and 



Q = 2uv — t (x, y) , 



therefore it follows that u = and i' = and therefore that 

 g (z) = 0. Moreover, since 



P'xo+?Q'xo = Oand P'yo+i'QVo = 



