Mr. J. W. L. Glaisher on the Problem of the Eight Queens. 463 



To obtain from the original and the rotation solutions the 

 corresponding set of reflexion solutions, we might, of course, 

 merely reflect each as it stands ; but a more expeditious process 

 is to replace every letter by its conjugate throughout, or every 

 suffix by its complement throughout. By either of these latter 

 methods we get all the reflexions : thus the reflexions answering 

 to (i)-(xii) are either b^m s d 7 c G k 5 a n ff m , &c, or c l2 n 6 e 7 b s h 9 a 3 f 4 , &c. 

 It may be here remarked that in all the solutions in this 

 paper the constituents are written in the order of the rows or 

 the columns in which they occur : thus in (i) c 2 belongs to the 

 top row, 7? 8 to the second, and so on ; in (vii) n Q belongs to the 

 right-hand column, b l<2 to the column next it, and so on. 



It is convenient to introduce the following definitions : — 



An un symmetrical solution is one which involves no vice 

 square \ thus an unsymmetrical solution gives rise to seven more 

 solutions, viz. 3 by rotation and 4 more by reflexion. 



A symmetrical solution is one which remains unaltered when 

 the board is turned through 180°. 



A quasi- symmetrical solution is one which is not symmetrical, 

 but which involves one or more vice squares. 



A pair of conjugate solutions are such that when the board is 

 turned through 180° each reproduces the other; two solutions 

 cannot, therefore, be conjugate unless they are quasi-symmetrical. 



Thus (i) and (ii) are quasi- symmetrical, as although they 

 respectively involve the vice squares n 8 and m 6 they are not sym- 

 metrical ; (iii) and (iv) are conjugate solutions, as the first letter 

 in (iii) and the last in (iv) are conjugates and their suffixes are 

 complementary, the second letter in (iii) and last but one in 

 (iv) are conjugates and their suffixes complementary &c. ; (v) and 

 (vi) are symmetrical, as in both of them the first and last letters, 

 the second and last but one letters &c. are conjugates, and the 

 first and last suffixes, the second and last but one, &c. are com- 

 plementary. The two four- solutions and the two penultimate 

 five-solutions, previously found, are doubly symmetrical. 



We now come to the chessboard of 64 squares, 



d 3 b 4 

 h fe 



G h 7 



Pi 



P9 



d~ 



/a 



m 



10 



a 7 

 /lO 



A 5 

 96 



>6 ?7 *8 



9% 



10 



d u 6 12 



9\o 



/jo dm b 



Jl 



12 



13 

 "14 



(h 



10 



II 

 9\2 



