440 Oa the Integral of the general Equation in Differences. 



deducible from the circumstance that u n and v will be represented 

 by the same determinant turned round into a new position. 

 But by means of our general representation for the case of any 

 number e of variables, we see that there is an analogous theorem 



which connects together 2 e different results, and which is not so 

 immediate a consequence of the theory of determinants. 



To make my meaning more clear, if we suppose the four fol- 

 lowing systems of equations, in each of which m > jjl, n > v, 



\/t, v / 



U M> „=S( ' )v m , v [v m>0 =l, V m+e) o=0, V m _ e ,-/=0, V m+e} Lf=0]j 



(777 LL \ 

 ' )Wn,n [Wo,»=l, W 0)n+f =0, W-. etn - f =0, «>_*,„+/= 0], 



(0H )V — Zi[ )<tim,n \_ a >7n,n— ±) w m+e, n— f = U, ^>m-e, n+f~ 0, <W m + e; „ + y=0j , 



we shall have u m , n — v , n = w m , o = &>o, o- 



The theorem u n = v above given, when the equation of differ- 

 ences is of the second order, expresses the well-known theorem 

 that the cumulant [#, b, c } . . . A, k, T] (the denominator of the 



contained fraction , z — , - — , . . . ^ — , T ) is the same as the 



aV b+' c + ' k+ I 



cumulant [/, k, h, . . .c, b, a~\ . 



There is no known property either of cumulants of this kind 

 or those of the higher orders, nor can there be any found, but 

 what does and must flow as an immediate consequence from the 

 representation of the linear-difference integral above given. For 

 instance, the law of formation of the above cumulant by reject- 

 ing consecutive pairs of terms becomes intuitive ; for to meet 

 this case we must write descending series of integers n, n v tz 2 , 

 . . . w w , 0, such that each difference between consecutive terms 

 n i} n i+l is always 1 or 2, and when the latter (m, n i+1 )=l. 



So more generally if we write u n =a n . u n _ l -\-u n _ ri we obtain 

 an analogous law for throwing out in every possible way groups 

 of r consecutive terms in order to express u n in terms of a n) a n -i, 

 a n - 2 , - - • a o So, toOj if we write u n =-u n _ x -f b n . w M _ M , we obtain 

 Binet's law of " discontiguous " products given in his long memoir 

 on the subject published in the Memoir es of the Institute, — 



* Or, more simply and rather more accurately, in place of the three equa- 

 tions within the bracket it is better to write u p> q =0 when jo or q or each 

 of them is negative, and so analogously for the cases following: — 

 v P) 7 =0 when m—p or q or each of them is negative, 

 W P, '/ = when m ox n — q or each of them is negative, 

 (o Pt 7 = when m—p or n — q of each of them is negative. 



