308 Dr. Roget on the Problem of the Knight's Move at Chess. 



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a^x ^p and proceeding either to the 



right or left, as the case may be, the circuit of each system 

 must be gone over in succession, according to that order: 

 beginning with that system to which the initial square be- 

 longs, and ending with that of the terminal square ; taking 

 care, however, for the reason above given, to avoid ending 

 the intermediate circuits at a corner square. It will be ad- 

 visable also to avoid ending these circuits at a square situated 

 on the borders of the board, for they will not always admit 

 of a transition being made from them to the next system into 

 which we have to enter. 



2. If both the given initial and terminal squares belong to 

 the same system, omit, while going over that system, the tei'- 

 minal square, and also one in immediate connexion with it *; 

 and fixing on some square in another system, which may be 

 connected with it, proceed as before, taking care to end at 

 this last-mentioned square ; whence, when ihe rest of the en- 

 tire circuits are completed, the two omitted squares may be 

 attained, and the conditions of the problem satisfied. 



3. If the initial and terminal squai'es belong, both of them, 

 either to systems denoted by consonants, or to systems de- 

 noted by vowels, the same course with that just described 

 must be pursued when the system to which the terminal 

 square belongs is gone over, and with the same ultimate re- 

 sult. 



Examples of each of these cases are given in the three 

 lower diagrams, the path of the knight in his course over the 

 board being traced by oblique lines joining the centres of the 

 squares he traverses ; the commencement and end of each 

 course, which are supposed to be previously given, being 

 marked by a small circle. I have made the second an example 

 of a recurrent circuit, in order to show that this condition 

 adds no new difficulty, and makes no difference in the mode 

 of proceeding. 



In these examples, the given initial square is the same in 

 all of them, and belongs to the system L. In No. 1, the 

 terminal square belongs to the system A. Here, we first go 

 over the whole sixteen squares of system L ; thence, passing 

 over to system E, we traverse all the squares of that system. 

 We next enter system P, covering in succession all the 



* The omission of this second square is not absohitely np^^ssary, but 

 will generally be found to facilitate the subsequent operations. 



