224 Mr. H. M. Taylor on the Relative Values 



white or a black square. The number of squares attacked by 

 a bishop on any square of the chessboard 'is given by the fol- 

 lowing scheme : — 



a 



a 



a 



a 



a 





a 



b 



b 



b 



b 





a 



b 



c 



c 







a 



b 



c 



d 



















TM„™v^« ~? Number of such 



Bishop on N c Tecks positions of 



cnecKS. bishop. 



a 7i—l 4(?i— 1) 



b n+1 4(n—3) 



c ti + 3 4?n— 5) 



d Ti + 5 4(n—7) 



71 



If 7i be even, we have x lines of this scheme all obeying the 



n—1 

 same law ; if n be odd, we have — - — such lines, and another 



for the middle square of the board for which the number of 

 checks =2(ti— 1). 



Now m terms of the series 



( n _l)( w -l) + ( w + l)( n _3) + ( n + 3)( n _5) + & c . 



= n 2 -27i+l + n 2 -27i-1.3 + 7i 2 -27z-3.5 + &c. 



+ 7i 2 -27i-(2m-3)(2m-l) 

 = w (ri 2 -27z) + l-(1.3 + 3.5 + 5.7 + ... 



+ (2ro-3)(2m-l) 



=7,(n 2 -2,) + l-^- 3 X 2? - 6 1 )^ + 1 ) + 3 . 



Put m= -, then the numerator of the chance-fraction 



2' 



=4{|(» 2 -2») + l-K«-3)(»-l)(» + l)-i} 

 = J(«-l)(2n-l). 



