308 Dr. Roget on the Problem of the Knight's Move at Chess* 
L^e 
av^_^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 ter- 
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 the 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 squares 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 absolutely necessary, but 
will generally be found to facilitate the subsequent operations. 
