OF FINITE DIFFERENCES. 205 



This value, in wliich G and r are arbitrary, satisfies (1) and 

 (2), and in consequence of the linearity of these equations they 

 will be satisfied by a sum of similar values, and we shall thus 

 have a more general solution, viz. 



sn 



If in this we put z = 0, we have 



Now, by (3), this is to be equal to p fora?=l, and to for 



the a 2 values of x, 2.3 a 1. There are thus a 1 



conditions for (12) to fulfil, and therefore we have, extending 

 the summation S from r = 1 to r = a 1, the following system 

 of equations : 



TT . a I 



i sn + -an ^-ir 



Q=C t sin + + CL, sin 2 TT 



^ ^ 



= 



(13). 



From these a I equations we have to determine the a I 

 quantities C t ... C a _ v . In order to do this, multiply the first 



T T 



equation by sin - TT, the second by sin 2 - TT, and so on, 



(r being an integer less than a), and add. Then, as may be 

 easily shown, the coefficient of every one of the quantities (7, 

 except (7 r , will in the resulting sum be equal to zero, while 



that of C r will be-. Consequently (13) is equivalent to the 

 2 



system of equations included in the general formula 



and consequently (11) becomes 



1 ^ + 1 / ^i\ 2 A. y / A. \ * 



ty = - (<nq\ * [*-} 2** 1 sin- TT sin TT cos -TT) (15), 



a \qJ a a \ a J 



which is the required probability. 



