[ '53 1 



Lemma. 

 If m and n reprefent any affirmative whole numbers : then 



-^ : n — I. n — 2. n — 3. 

 — : n — 2. n — 3. n — 4. 

 + : «^3. n^. n^. 



n — m — I X ^ 



n — m X m 

 H-HTz+l X m. m — I 



=:: o 



I 



J 



+ :« — /« + i. «— 7W + 2 - .- - « — ivt — i^i 



Where i, w, m. m — r, &c. are formed by the law of the 

 2 

 coefficients of a binomial raifed to the w'*. power. The nvimber 



of terms = ;?z-f- I. 



Demonstration. Let the terms of the rainexed table repre- 

 fent the different exprcffions for the above quantity, according to 

 the different values of m and n. 



Vol. Vil. 



U 



Then 



