458" 



DEMoffSTRATioN. — The first differences of the " po\ver3 

 of the Binomials are derived by taking away the first members 



th 

 of those powers, for this is to subduct the " powers of the 



th 

 next lesser Binomials. The second ditferences of the " 



powers of the Binomials (or the differences of the first differ- 

 ences) will be ecjual to the sums of the separate differences of 

 the separate members of the first differences, which members 

 being the powers of terms in an arithmeMcal series, the in- 

 dices of which powers are " — ■!, " — 2, " — 3, &c. the first dif- 

 ferences of these will be known from the first differences of 



the ^' powers : For, \i p'^—ff^—ydq^ +C(rq"^ +&c. then 

 willp^ — ryr '-[';— \)dqf ^+Cd'qy -h&c. ; and p^ — 



z 

 m n m Q m . .\, 



qT ={';—2)dqr +Cd'qy +&C. 



3 



But there are wanting none except the highest members of 



the expressions for the first differences, in order to obtain the 



highest members of the expressions for the second differences, 



th 

 and none except the highest members of the n-^ differences 



th 

 in order to obtain the highest members of the n differences, 



for the differences of the highest members contain higher 



powers than the differences of the lower members. 



Since then, the highest members of the first differences 

 (h 

 of the " powers are "d multiplied by the powers of num- 

 bers 



