Calculus of Finite differences. 1 29 



. n/^-2 » /„ i\«-2 , 11.11— I, \ m ~ 2 

 + QK» — * (» — I ) H — — (m — 2) 



+ S |w' - n (« - 1)' + "'j^ ] (« - 2) 1 

 (»l - 3) 1 . . ., &c.\ 



».» — !.«-— 2 



1 2 3 



— 2Y> 



+ T (n° -n{n- 1)° + n ' 11 * (n - 2) 1 

 (n-3)° "V. 



n . » ,— 1 * n -f- 2 



(III.) 



1 2 3 



The quantities between the brackets in the common notation 



are thus written: A" 0™, A" 0" i_ \ A" O m ~ 2 , which are each 



cypher when n is greater than m ; hence A w x™ — when 

 n > m. 



The theorem A" 0"' = 0, or its development, 

 n — n [n — 1) H — — (m — 2) 



N . 11 — 1 . 71 — 2 . „ .* 



rrr - ( "- 3) =0 •- 



may be proved by the common principles of algebra as follows : 

 Assume x = t — 1, raising both sides of this equivalence to 

 the nth power, we have 



n n n-\ , 11 . 11 — \ n -2 11.71— 1 .11 — 2 n -3 „ 

 x=z-nz + 12 z — Z , &C. 



Let the first derivative of this equivalence be taken, or, in 

 other words, let the equivalence x = z — 1 be raised to the 

 (« — 1) power, and then multiplied by «, we thus find 



«_1 „_i w _2 ii.li — \.n—2 n ~' 6 a 

 nx — nz — n(u—])z H z , &c. 



Multiplying this series by the equivalence x + 1 = z, we 

 obtain 



nx n + nx n - l = iiz n -n(n-\)z , '- l + } -^^(n-2)z"- 2 i &c. 



Again, taking the first derivative of this series, and multiply- 

 ing by x + l=z, there results the equivalence 



Phil. Mag. S. 3. Vol. 19. No. 122. Aug. 1841. K 



