Mr. J. W. L. Glaisher on the Problem of the Eight Queens, 465 



and the latter to the two, 



gjfc&ii&ffltymaft b 14 p 7 f 8 g l2 e 9 e 6 a 3 k 5 . 



Each of these 28 solutions gives rise to another by replacing the 

 letters by their conjugates, or the suffixes by their complements : 

 there are thus altogether 56 penultimate solutions. 



Of the antepenultimate solutions involving e 3 as leading square, 

 the first is symmetrical, the second and third are conjugate, and 

 the other three are quasi-symmetrical — two involving one vice 

 square each, and the remaining one two vice squares. By rota- 

 tion they only give rise to 4 more solutions, 



n i(i e n k 7 f 12 g 4 h 9 d 5 m 6 n l0 c l2 d l3 k 5 b 8 h 9 a 3 f 4 

 ttio e i 1^13^8 ^A^3^6 d\ 3 h 9 m§g X2 a 7 b 4 n 8 k 5 . 



Each of these 10 solutions gives rise to another by replacing the 

 letters by their conjugates, or the suffixes by their complements ; 

 there are thus altogether 20 antepenultimate solutions. On the 

 whole, therefore, we have 92 solutions, of which 16 are ultimate, 

 56 penultimate, and 20 antepenultimate. The number 92 is 

 the same as that found by Gauss. 



Every unsymmetrical solution, as before remarked, forms one 

 of a group of eight, connected together by rotation or reflexion ; 

 and it is natural to inquire how many essentially different solu- 

 tions the problem admits of. By the essentially different solu- 

 tions (or, say, the type-solutions or stem-solutions) are meant 

 all the solutions, if we regard a solution and all that can be de- 

 rived from it by rotation and reflexion merely as one solution. 

 The following will be found to include all the stem-solutions : — 



a l5 n % e 9 k b f l(i d 7 b±m 5 c 2 e 5 g 8 k n d 7 m 6 a 13 f l2 e 3 g 6 b 4 k n h 5 c l2 f i0 d 13 

 «i5%c \^e 7 gji 9 d b m^ c 2 g 6 k 9 f 4 d 7 e l3 b 12 h n e 3 g 6 n l0 a 7 h 5 c 12 m 8 d 13 



c 2 g 6 k 9 a 7 h 5 e l3 b l2 m 10 e 3 k 7 b 4 c 8 g l2 m 6 a l3 h nf 



c%k 7 d 3 g x 0^8^13^9/1 2 



c 2 k 7 n l0 b 6 h 5 d 9 a l3 f l2 



c 2 n 8 e 7 k n h 5 d 9 b 12 m l0 



c 2 n 8 a 5 e 9 g l2 b l0 p 7 h n 



the two in the first column being ultimate, the seven in the 

 second penultimate, and the three in the third antepenultimate. 

 There is but one symmetrical solution (viz. the first in the third. 

 column) ; and the number 92 may be accounted for as follows : 

 if none of the twelve stem-solutions were symmetrical there 

 would be 8x12 = 96 solutions; but one, being symmetrical, 

 gives rise by rotation and reflexion to only three new solutions 

 Phil. Mag. S. 4. Vol. 48. No. 320. Dec. 1874. 2 II 



