48 PROCEEDINGS OF THE AMERICAN ACADEMY 



and since the #'s are nilfactorial together, i. e. since &i & 2 — #2 l ^i 

 = 0, etc., therefore 



etc., etc. 



Consequently, if f{&) is any rational integral function of 6 of 

 which k is the term independent of 6, and if f{&) = f {&) — k, 

 then 



f(0) =f(6) + k =/(*0 + f(& 2 ) + /(*,) + •••• + 7c. 



Let [r, s] denote the matrix whose constituents are all zero 

 except that in the r-th row and s-th column, which is unity. We 

 have then 



*i = 9x ti m r [?, r] + ^V [r, r + 1], 



#2 = ff 2 2i" [m + r, m + r] + S/^ 1 [m + r, m + r + 1], 



#3 = #3 Sir [m+w+r, m+n+r] + 2/7 * [wi + w + r, m + rc+r+1], 

 etc., etc. 



By the laws of multiplication of the symhols [r, s], viz. 

 [r, s] [*, f] = [r, t], [r, s] [*, v] = (5 =t *)> 



it follows that 



xV - r/i 2 XTr [*. '] + 2 * SiV 1 [r, r + 1] + SiT 1 [r, r + 2], 



#i 3 - ?i 3 S™ [r, r] + 3 fl« S/V 1 [r, r + 1] + 3 ffl S.V" [r, r + 2] 



+ Srr 3 [r, r + 3], 



xV = ^S * [r, r]+^*" XT' ['1 •+ 1 ]+2^^"" f [^ y + 2 ] 



1 d m_1 



+ •••• + (^^iyt^-^[l,m]. 



Therefore, if f'(ff), /"(#)> e ^c., denote respectively the first, 

 second, etc., differential coefficients of f(g) with respect to g, 

 then 



