Calculus of Finite Differences. 127 



It is manifest that the coefficients A, B, C,...M, N are the same 

 in the successive developments, as being functions solely of 

 a, /3, y, 8 . . . X and m ; making these substitutions in (I.), this 

 series may be now thus written : — 



V = {k m + Af" 1 + Bf 2 Uk + N} 



- n {/c" n + A IT" 1 + B k' m - 2 M H + N} 



+ n ' n ~ 1 {r+Ar-'+BF- 2 MF+N} 



1 2 



n .71— 1 . n — 2 



{F^&c.&c.}. 



1 23 



Now, putting for #, k", F', their values (Jc—h) (k—2h), &c, 

 or, as k is an arbitrary multiple of /z, let £ = u h, and adding 

 the terms of the above series vertically, we find 



= /z -S ^ — « (W — 1) H r-g — (« — 2) 



(m - 3) m . . ., &c."| 



n .11 — 1 . w — 2 



12 3 



+ Ar-^ii'-^nCtt-ir-^—^Cw-S)"" 1 

 {u- 3) m -\.Mc.\ 



n . n — 1 . n — 2 , „sto— 1 



1 2 3 



JH-2 



4- Br- 2 izz re - 2 - W («- l) w " 2 + '^Ip 1 («-* 

 ( W -3) m_2 ...&cA 



w . w — 1 . M — 2 



12 3 



+ M hi u - »(« - 1) + n, ^ g * (« - 2)...&c.T 



+ N/z°|w -«(w-l)° + ^^(zz--2) ...&cA. 



Now, by a known theorem in the calculus of differences, the 

 quantities between the brackets are the nth differences of the 

 with, (m — 1) and lower powers of the number (u — n). 

 Hence 



= Jim A (?Z — w) + A A A (?Z — ft) 



+ MA A" (w - n) + N#>A n (« - *)°J 

 and as ?« by hypothesis is less than «, the wth differences of 

 («—«)» ( M _„) m - 1 are eac h equal to zero ; hence, as A, B, C, 



