NOTE B. ARTS. 53, 54. 



Lemma. If, in any region, a circuit* or system of circuits A is 

 reconcileable with a system B, and also with a system (7, the systems 

 B and G are reconcileable. 



For it is possible to connect, in the first place A and B, in the 

 second A and (7, by continuous surfaces lying wholly within the region 

 and completely bounded by the lines A, B and A, C respectively. These 

 two surfaces adjoin one another along the lines A, and if these lines be 

 obliterated we obtain a continuous surface bounded by B and (7, which 

 are therefore reconcileable. 



Suppose now that we have in any region a system of n independent 

 circuits a l} a 2 , ... a n , such that every other circuit drawn in the 

 region is reconcileable with one or more of these. We shall prove that 

 any other set of n independent circuits b lt b a , ... b n which can be drawn 

 in the region possess the same property. 



Let x denote any other circuit drawn in the region. By hypothesis 

 6 t is reconcileable with one or more of the system a lt a g , ... a n , say with 

 a lt combined (if necessary) with others of the system; and x is also 

 reconcileable with one or more of the system a lt a a , .... Firstly let us 

 suppose that a l is included in the set with which x is reconcileable. 

 We have then a^ reconcileable with one or more of the system 

 b l , a 2 , 3 , ... a M , and also with one or more of the system x, 2 , a 3 , ... a n . 

 Hence, by the above Lemma, it must be possible to form a mutually 

 reconcileable set from the system x, b^ a a , a z , .. Further, of this 

 set x must be one ; for otherwise we should have b l reconcileable with 

 one or more of the system 3 , er&amp;lt; 3 , ... a nt and also with a } (combined, it 

 may be, with others of the system). Hence, by the Lemma, it would be 

 possible to form a reconcileable set from the system a l9 a 2 , ... a n , con 

 trary to hypothesis. Hence x must be reconcileable with one or more of 

 the system b l , a 2 , 3 , ... a n . If a t be not included in the set of a s with 

 which x is reconcileable this result is obvious. Hence, in any case, we 



* Every circuit spoken of in tin s note is supposed to be simple, and non- 

 evanescible. 



