156 



the constant members of the preceding order of differences 



th 

 are destroyed, therefore in the n differences the first terms 



alone will I'emain, and the indices being =-n — n the powers 



Ih th 



will be equal to unit. Therefore the n differences of the ?j 

 powers are reduced to the constant quantity nd. [ii — \)d. 



{n—2)d.Scc..3.d.2dAd=1.2.3.&:c..{7i—2).{>i—l).7i x d" 



Sir Isaac Newton's Binomial Theorem may be deduced 

 from a simple algebraical Equation, derived from those Co- 



rollaries and from what has been shewn of the form of x + qy 

 by substituting merely the terms of an arithmetical series for 

 q, viz, 



T + q ^"r—x'r + bqa:^ + cqi^ + dq.v^ + &C. + ^q^^ + &c. 



,j H- ^ ^ = AT'^^ + ^4..r^ +c.4A,-r ' j^dAx~ +&:c. + /.4.i^ +&c. 



x + S'^—x^ + o.Sx'' +c.3.i'^ '' + do.x'7 -\-kc. + t.3.x'^ +8ic. 



.T+2 '^=:.t^ + &.2Ar'- ^-c.'i.x^ j^d.l.x'^ + &c. + f .2.jc~ -f&c. 



th 



Here the Ji powers of the natural numbers multiplied by 



^ will be the co-efficients of a-^""' and since the terms which 



contained 



