Mr. J. W. L. Glaisher on the Problem of the Eight Queens. 459 



a \ c i e s 9i 



ffti 



d 3 b, 

 h 



h *, 



(i) 



(or, say, the five-sol utioDs) we may proceed as follows. To ob- 

 tain those that involve a 9 we have only to append a 9 to every 

 four-solution that does not involve an a : it happens that neither 

 of the four-solutions does involve an a, so that CQe b d s b 6 a 9 and 

 e 3 b, 2 c 6 d 6 a 9 are the two solutions that involve a 9 . Evidently the 

 two that involve a l are found by replacing each suffix by its 

 complement to 10, and are therefore c 8 e 5 d 7 b 4 a 1 and e 7 h B c 4 d^a^ 

 To obtain the corresponding solutions involving h h and k b we 

 have only to " reflect " the four solutions we have obtained (i. e. 

 replace in them the rth square from the right in any row by the 

 rth square from the left) ; we thus get 



ff 4 a Q e~b 6 h 5 , e 3 g 6 b 4 a 7 h 5 , f 4 a 3 d 7 c 6 k b , d 3 f G c 4 a 7 Jc b . 



These are the only solutions in which corner squares are in- 

 volved (or, say, the ultimate solutions) ; and in developing the 

 determinant we may replace the four corner squares by zeros. 

 The actual development in effect is 



c 2 e s 9\ • —H-\' ' C h 9 6 + e -- 



e o 96 



ch b A a 



h 



fe d 7 h 



do a. 



k 



. b. 



<4 «-, 



feh 



(2) 



the first term on the right-hand side giving all the solutions 

 involving c 2 , and the second those involving e 3 . It is clear that 

 if we have obtained the solutions that involve c 2 , we can derive 

 those that involve g 6 , b 8 , and f 4 by merely turning the board 

 through 90°, 180°, and 270°; while those involving^, c 8 , f 6f 

 and b q are obtained at once by reflexion. The first term of the 

 right-hand member in (2) merely consists of c 2 multiplied by 



its minor, 



the 



constituents involving c 



and the suffix 2 beim 



replaced by zeros ; the third term consists of e 3 and its minor, 

 the constituents involving e and 3, and also g 6 , b 8 , f 4 , c 8 , f 6 , £ 2 , 

 being put equal to zero. This term therefore takes the form 

 in (2) and is seen to vanish. The former term 



■ c e 



2'- 5 



A h 



d~ . 



+ C z96 



^3 "5 ' 



i • * 7 h 



= c *9&bfi h *'> 



