LAPLACE. 529 



wc may write this result thus, 



Avhere i^(^) and /(r) are functions of t and r respectively, which 



are at present undetermined. By supposing that the term in /(t) 



vv'hich is independent of t is included in F{t), we may write the 



result thus, 



„ = %IO±ltM (4)_ 



1 — pt — qr 



Thus either (3) or (4) may be taken as the general solution of 

 the equation (1) in Finite Differences; and this general solution 

 involves two arbitrary functions which must be determined by 

 sjDecial considerations. We proceed to determine these functions 

 in the present case, taking the form (4) which will be the most 

 convenient. 



Now A loses if B first makes up his set, so that <^ {x, 0) = 

 for every value of x from unity upwards, and (/> (0, 0) does not 

 occur, that is it may also be considered zero. But from (4) it 

 follows that (j) (x, 0) is the coefficient of f in the development 



of ^^^ ; therefore v (t) = 0. 



Again, A wins if he first makes up his set, so that cj) (0, y) = 1 

 for every value of i/ from unity upwards. But from (4) it follows 



that (j) (0, y) is the coefficient of t^ in the development of r-^ 



so that 



Tyjr (t) _ T 



qr 



therefore 



Thus finally 



{l-T){l-pt-qT) 



Now <^{x,y) is the coefficient of fr' in the development of w. 

 First expand according to powers of t ; thus we obtain for the 



34 



