155 



CoK. I. If instead of o, p^ q, r, &c. the terms of the 

 arithmetical series be supposed x+o, x+p, x+q, x+r, &q. 



th th 



then the first members of the n differences of the ^ powers 



will be ?rf (^— l)rf. (^— 2)J. &c. {^-l^\)d multiplied by the 



th 

 7 — 11 powers of Binomials in the arithmetical progression 



x + o, x+p, x + q, Sec. and if the terms in each difference are 



made to proceed according to the powers of at, the highest 



th 

 member of each of the n differences will be constant, and 



equal to '^d. ("— l)rf. ("— 2)c^ &c. (^— »— 1) dx^ '• 



Cor. II. The second differences of the Squares, the third 



th 

 differences of the Cubes, and in general the n differences of the 

 th 



n powers of numbers in arithmetical progression Avill be con- 

 stant and equal to the product of the digits from the Index 



th 

 to unity, multiplied by the n power of the common difference 



n 



of the arithmetical series, or =1.2.3. &c, nxd. For, each 

 order of differences being made up of the separate differences 

 of the separate members of the preceding order, and the in- 

 dices of the members diminished by unit being the highest 

 indices in the differences of those members, when those in- 

 dices are equal to cypher, the members will be constant in 

 which cypher is the Index of the terms. In each difference 

 VOL. XI. y the 



