174 UNICURSAL PROBLEMS [CH. IX. 



if this random coupling produces anywhere an isolated loop, L, 

 then where it touches some other loop, M, say at the node P, 

 unfasten the four hooks there (viz. two of the loop L and two 

 of the loop M) and re-couple them in any other order: then 

 the loop L will become a part of the loop if. In this way, 

 by altering the couplings, we can transform gradually all the 

 separate loops into parts of only one loop. 



For example, take the case of three isles, A, B, G, each 

 connected with both the others by two bridges. The most 

 unfavourable way of re-coupling the ends at A, B, would be 

 to make ABA, AGA, and BGB separate loops. The loops 

 ABA and AGA are separate and touch at A; hence we should 

 re-couple the hooks at J. so as to combine ABA and AGA into 



A 



one loop ABAGA. Similarly, by re-arranging the couplings 

 of the four hooks at B, we can combine the loop BGB with 

 ABAGA and thus make only one loop. 



I infer from Euler's language that he had attempted to 

 solve the problem of giving a practical rule which would 

 enable one to describe such a figure unicursally without 

 knowledge of its form, but that in this he was unsuccessful. 

 He however added that any geometrical figure can be de- 

 scribed completely in a single route provided each part of it 

 is described twice and only twice, for, if we suppose that every 

 branch is duplicated, there will be no odd nodes and the figure 

 is unicursal. In this case any figure can be described com- 

 pletely without knowing its form : rules to effect this are 

 given below. 



Third. A figure which has two and only two odd nodes can 

 be described wnicursally by a point which starts from one of the 

 odd nodes and finishes at the other odd node. 



