312 



G. H. KNIBBS. 



Fig.33 a. be. . Rg34 



[o<sW$> ( c > K^s^i 



(d) /-■ 



=2= (e) | 3 

 Fig. 32 a toe V% 



distance from the edge, 1 it will be found that the band is divided 

 into twisted interlocked rings, one of which is identical with the 

 original in length and twist, and the other is twice the length and 

 is more twisted. A cut along the centre line from O returns into 

 itself after one circuit, and does not divide the ring into two 

 portions. A cut like AB makes it a simply-connected 2 surface. 

 Fig. 33 (c) will exhibit similar peculiar features. A closed curve 

 in an ellipsoidal surface will necessarily separate it into two simply- 

 connected surfaces, whatever its position ; in a tore or anchor-ring 

 it may or may not do so, according to the position of the cut. 



A surface like that in Figs. 33 (a) or (c) cannot, by extension, 

 enclose space; if however the twist is 2ti and it first interpenetrate 

 its own surface, then by extension it can. Fig. 34 is a symmetrical 



1 The cut is shewn by the dotted line. If traced the heavy dots denote 

 the visible line, the light ones that on the remote side. 



2 That is one which every possible cross-cut from boundary to boundary 

 will divide it into two parts. 



