1.^7 



I 



th 



contained the first, second, tliird, Sec. to the n — 1 powers of 



th 

 the natural numbers will not be found in the ii diffei'ences 



th 

 of those series, (the n differences of powers whose Index is 



less than n, being differences of common differences,) therc- 

 fore the first members in all the n differences of the " powers 



ih 



of iH-4, x + 3, x + 2. Sec. Avill be = / multiplied by the n dif- 



ferences of the ?j powers of natural numbers x xr or (by 

 Cor. 2.) =t X 1.2.3. 8cc. nxA:"~". But (by Cor. 1.) those 

 common first members =".(" — 1).(" — 2).&c.("— n — i)*:^ • 



therefore < X i.2.3.&c.nx:i'7 "_- (-_i).(»_2).&c.("— n— 1) 



Wi- 



tt 



yxr ".•.< = ^.(^— 1).(^— 2)..&c.(^— n— 1). 

 1. 2. 3.. &c. n 



n 1 a m „ g 



Therefore x + q'r= XT + ^qx7 +;'.(^— i) g3:7 --+&c.-f 



1.2 



" m, — n 



';.{>^—l).Scc.{^~n—l)q x'- +Scc. Q. E. D, 



1. 2.. &c. u 



Note. In proving the Binomial Theorem above, I have 

 equated the two expressions for the common first members of 



th th 



all the n differences of the ^ powers. For, the differences 

 themselves are indentical, being changed only in form without 



y 2 altering 



