238 NOTES. 



can, in the system a lt 3 , ... a n , replace a } by b lt and still obtain a 

 system of n circuits with one or more of which every other circuit drawn 

 in the region is reconcileable. 



In particular, b 2 is reconcileable with one or more of the system 

 b lt a 2 , a 3 , ... a n ; and it is, by hypothesis, not reconcileable with b l 

 alone. If then a a be one of the set with which it is reconcileable, 

 it follows by exactly the same argument as before that we may replace 

 a 2 by 6 2 , so that any circuit drawn in the region is reconcileable with 

 one or more of the system b lt b 2 , 3 , ... a n . 



This process of replacing a s by 5 s may be continued, until we 

 finally arrive at the result that every circuit drawn in the region is 

 reconcileable with one or more of the system b lt b 2 , ... b n . 



The order of connection of a region is therefore a perfectly definite 

 number. For let there be two sets, a lt a. 2 , ... a m , and b lt b. 2 , ... b n , of 

 independent circuits, such that any other circuit drawn in the region is 

 reconcileable with one or more of either set ; and if possible let m, n be 

 unequal, say n &amp;gt; m. The above argument shews that every circuit 

 drawn in the region is reconcileable with one or more of the system 

 &i ^2? & w contrary to the supposition that there are n - m independ 

 ent circuits b m+ lt ... b n which are not reconcileable. 



It is possible to draw a barrier meeting any one of the circuits 

 a l9 er- 2 , ... a n in one point only. For if every barrier through a point P 

 of one of these circuits meets the circuit again, it must be possible to 

 connect P with all other points of the circuit by a continuous series of 

 lines drawn each on a barrier, and therefore to fill up the circuit by a 

 continuous surface lying wholly within the region, contrary to the sup 

 positions that the circuit is simple and non-evanescible. 



Further, it is possible to draw the barrier so as not to meet any of 

 the remaining circuits. For if every barrier which meets one of the 

 circuits, say a lt necessarily intersects one at least of a certain set a of 

 the remaining circuits, then, if we consider the infinite variety of 

 barriers which can be so drawn, it appears that we can connect the 

 various points of a } with those of a by a continuous web of lines drawn 

 each on a barrier, and therefore we can connect a l and a by a continuous 

 surface lying wholly within the region. That is, j and a are reconcile 

 able contrary to hypothesis. 



The above demonstrations are slightly modified from Rdemann* and 

 Kb nigsberger-t. 



* /. c. Art. 63. 



t Theorie der ell ipti when Funt tioiien, c. 5. 



