ESSAYS 



ON 



POWERS AND THEIR DIFFERENCES. 



BY FRANCIS BURKE, Esq. S(c. lie. 



THE SECOND ALGEBRAICAL ESSAY. 



On finding, per Saltum, the several Orders of Differences. 



INTRODUCTION. 



jL he Formulas for finding, per Saltum, the several orders of 

 differences of quantities in a series, have been usually de- 

 duced from a repeated algebraical subduction. Thus, if the 

 successive quantities are a, b, c, d, e, &c. the first differences 

 area — b, b—c, c — d, d — e, &c. ; and the second differences, 

 or the differences of the first differences, are, a — 2b + c, 

 b — 2c+ci, c—2d+e, &c. ; and the third differenc es u — .%+ 

 3c — d, b — 3cH-3f/ — e, &c.;and, the coefficients being produced 



n th 



like the unciae of the powers of (1 — x), the n differences are 



Q— ?tfc+ »^»-'V " '"—')''— 2) ,/-i.A-r^ b Hr4- "(''— ') d n(»—l ■. (;!— 2)^ 



1.2 —1.2.3 ~r72~ 12 . 3 



m m m m 



+&C, &c. And if the quantities are a , 6 , c , rf , &c. 

 VOL. XI. z the 



