Ross — Verb-Functions, 51 

 Then, by operative division, 



tiP' + tA^' + tA(^' . . . 



Hence, 



ti = — (I, = — tihi — ^2, ^3 = — ^2<^l ~ ^1^2 ^3 j 



and, generally, 



+ t^^ki + t^_2j\ . . . toC^.o + tibr_i + = 0. 



I^ow, let us denote the successive coefficients in the ordinary 

 algebraic expansion of the m^^ power of a multinomial by 



{m)2, {m\, . . . ; 



so that 



Then it will be found by reductions that 



{m)i + 2m) = 0, {m2) + (2m\ + i(- 3w) = 0, 



Ws + 2/w)i (2m)2 + i (- 3»i)3 (3m)i + i (- 4^03 = 0> 

 and generally, 



{m\ + 1(- 2?/0i (2^0»-i + i(- 3»02 (3we)^2 . . • (- + 1) ^^0^ = 0- 



And this is true, if be a fraction such as jS'ow the coefficients 



n 



a, h, c, . . . employed above may be written 



with the proper subscripts. Hence the result of (1) may be written 

 (in reversed order), 



