TRANSACTIONS OF THE SECTIONS. 



23 



The above t-wo cases, which are the simplest, will suffice to show the method 

 pursued by the author. In the case of the bishop, n being even, the numerator of 

 the chance fraction is equal to twice the sum of the first ^n terms of the series 



(n-l)(n-l) + (7i + lXn-3) + (n + S)(n-5)+&c. = l)i(>i-l)(2n-l). 

 The results for the cases of n even are given In the following Table : — 



Chance of the Mng being in check. 

 For board of n- squares. 



For board of 64 squares. 



Knight 

 Bishop . 



Simple 

 check. 



«'*(«+ 1) 

 2 2«-l 



Rook .,.,,, 

 Queen ...,,, 

 Two bishops 

 Two rooks . 



3 n{n+l) 

 2 



2 5n-l 



3 »(n+l) 

 24ws— 9«+2 

 3 n(n'-2) 

 2(2m'~2h-1) 

 («+l)(«^-2) 



Safe 

 check. 



8(91^2) 

 iiXn+l) 



2 ( «-^2)(2«-3) 



3 M^(M+1) 



2(m-2) 

 w(w+l) 



2 (m-2)(.5«-3) 



3 n\n+l) 



It is to be remarked that the relative values of the knight, bishop, rook, and 

 queen are, according as we measure them by the chance of simple check or oif safe 

 check, on the ordinary chessboard in the ratio of 3, 5, 8, 13, and 12, 13, 24, 37 

 respectively ; while the values of the pieces in the same order, as given by Staunton 

 in the ' Chess-Players' Handbook,' are 3-05, 3-50, 5-48, and 994, the value of 

 the pawn being taken as unity. (The value of a pawn depends so much on the 

 fact that it is possible to convert it into a queen, that the method explained in the 

 paper does not appear applicable to it.) 



On Laplace's Process for determining an Arhitrary Constant in the Integration 

 of his Differential Equation for the Semidiurnal Tide, By Sir W. Thomson, 

 F,B.S., F.E.S,E, 



Oerural Integration of Laplace's Differential Equation of the Tides*. 

 By Sir W. Thomson, F.B.S., F.B.S.E. 



On the Integration of Linear Differential Equations with Rational Coefficients, 

 By Sir W. Thomson, F.B.S., F.B.S.E, 



On some Effects of Laplace's Theorii of Tides. 

 By Sir W. Thomson, F,B.S., F.B.S.E. 



* Published in exienso in the 'Philosophical Magazine' for November 1875, ser. 4, 

 vol. 1. p. 388. 



