1 28 Professor Booth on a new Theorem in the 



M, N, and li are finite, all the terms of the above series are 

 separately equal to zero, or 



V = 0. 



It is clear that, whatever be the nature of the numbers a, 6,c,...Z 

 and h, we may assume the arbitrary number h, so that u 

 may be an integer. 



The theorem A w (u — n) m — may be reduced to another 

 more simple, as follows : let u = x + n, then A n (u — n) m 



An ni > 

 x ; but . 



n . n - 1 . n - 2 . &c J 



1 2 3 



{Lacroix. torn. iii. p. 9.) 



developing these powers in the following manner : — 



(* + n) m --= n u + Pn m ~ l + Q n m ~ 2 Sn + T 



(.r + « - 1)" = (* - i) ? " + P (m - I) 7 " -1 

 + Q(w-1)"" 2 S(n-l) + T. 



P, Q, S, T being functions of x and m. Substituting these 

 values of (x + n) m , {x + n - \) n \ &c. in (II.), we find 



A\v m = / + P/- 1 + Qn"-*...Sn + T 



_ n {( „_ l )- + P ( w _i)— l + Q(n-l) m -*....S (n-l) + T 



71.11— 1 f. «\#» . r» / o\»' — 1 



+ j~2 — -j (w - 2) + P (w - 2) 



+ Q(» -2r _2 ...S(n-2) H 

 Now, adding vertically, we obtain 



+ Q (» - 2) OT_2 ...S (« - 2) + T, &c. &c. &c. 



'^'=-j?i — n (n — 1) h — — {n — 2) 



(«-3) W2 ..., &c."| 



A 



w . w — • 1 . n — 2 



1 2 3 



+ P^w — n(n — 1) + — — — (rc— 2) 



n . w — 1 . n — 2 



(«- s )™"'-' & 4 



