the general Equation in Differences. 439 



the arrangement of 8 l} S^, . . . S w becomes indifferent, and conse- 

 quently the value of u ni denned by the equation 



W„=(l)w n -1 + (2)w n _ 2 + . . . + (2> n _i, 



becomes the coefficient of t n in -= tt- — r^—^ r^-.-. as is 



1 — (l)t— (tyf ... — [i)t 



well known. 



The above rule may easily be extended to a linear equation in 



differences with any number of variables. Thus suppose, for 



greater simplicity, that we write 



~{x i d\ ra/=x— 1, x — 2, . . .01 



K V, V J L^=y-1, y-2, . . .0-J 



with the initial conditions u 0} = 1, u 6} f=0 wherever one or both of 

 e } f are negative units; then to find the value of m, n we must form 



all the possible descending series ' v 2 ' ' "' , subject 



Ln, n v Wg, . . . rc w , OJ 



only to the law that there is a descent either from m { to m i+l) or 

 from n t to n i+l , or at one and the same time from m i to m t+1 and 

 from n t to n i+1 . The value of u m> n then becomes 



y(m, m v m^ . . . m w , 0\ 

 \n, n Xi n 2 , . . . w w , 0' 



with the understanding that the term within the parenthesis is 

 to be read as meaning 



/m, Wj\ x /m v m 2 \ x /ro 2 , ™ 3 \ x /w* M , 0\ 



\w, n l ' \n v n 2 / \w 2 , n 3 / \rc w , 0/ 



And in like manner and under a similar form we obtain the 

 value of w Ml) „ 2) . . . n defined by the general equation 



n v ' 



ln defining the relations which connect one w with another, 

 we may suppose that (r, 5) means the coefficient of u s in the 

 equation 



u r = ^(r, s)u s [r > s, w =l, w_ e = 0] ; 



but we may also suppose that r, s means the coefficient of v r in 

 the equation 



v s =l(r, s)v r [r>s, v n =\, v n+e = 0] j 



the value of u , on the latter supposition, it is obvious, becomes 

 equal to that of u n on the former — a fact that is well known, and 



