On the Relative Values of the Pieces in Chess. 221 

 more simply. If, namely, we put 



r 



ee f 



r 



vvf cos e, 



_ z ee r /dx dx' ,dydif dz dz\ 

 ~ r\dt o¥ + dt~dt + Jt dt)' 



and regard U as a function of the six coordinates x, y, z, x r , 

 y f , z r , and V as a function of these six coordinates and their 

 differential-coefficients according to t, we can write 



dx dt\jdx f\ 



d ~di 



and in just the same manner can the other five force-compo- 

 nents be derived from the two functions U and Y by differen- 

 tiation. 



For the components of the force which is exerted upon a 

 galvanic current-element ds by a current-element ds' we get 

 from the simplified form of the fundamental equations the fol- 

 lowing expressions : — 



(d- d- 7 / N 



r r dx'\ % 



dx- C0Se --^)> 



d- d 1 - 



cii! ds ds{-L cos e- JL $£) ; 

 \dy ds as' / 



en 



.dy 



d l 



cii! ds ds'l JfL cos e — -^ 4~r X 

 \dz ds ds f J 



A - 1 



XXYII. On the Relative Values of the Pieces in Chess. By 

 H. M. Taylor, M.A., Fellow and Tutor of Trinity College, 

 Cambridge*. 



THE object of this paper is to ascertain the relative values 

 of the pieces on a chessboard. If a piece be placed on 

 a square of a chessboard, the number of squares it commands 

 depends in general on its position. If we calculate the ave- 

 rage number of squares which any particular piece commands 



* Communicated by the Author. 



