150 



CHAP. II. 



Of the Co-efficients of the Terms in the Binomial Theorem. 



From the following proposition two corollaries are easily 

 deduced, which La Croix has proved by expanding a com- 

 plex differential formula, which he has made to depend on the 

 co-efficients of the terms in the Binomial Theorem. (See his 

 Diff. Meth. vol. 3. pag. 7.) But as I shall shew an immediate 

 transition to the unciae of powers from those corollaries, I 

 shall simply demonstrate them by means of the following 

 indepencU'nt proposition. 



I shall first observe that the terms of an arithmetical se- 

 ries are usually represented as Binomials, whose first members 

 are the constant basis of the progression, and whose second 

 members are the variable multiples of the common differ- 

 ence: but we shall avoid any complex substitutions, if we 

 make the next lesser term of the arithmetical series the first 

 member, and the common difference to be the second member 

 of the Binomial. 



Prop. If o, -p, q. r, .?, &c. are the terms of an arithme- 

 tical series wiiose common difierence is d, then Avill p + d, 

 q + d, r + d, s + d, t + d, &c. be respectively equal to the cor- 

 responding terras of the series, o, p, q, r, s, «&c. If Ave take 



the differences of the " powers of the Binomials, and the dif- 

 ferences of liiese differences, or the second diflerences, &c. 



the 



