151 



■ <* ■ ih 



the first members of the n differences of the % powers of 



the Binomials, are the (" — n) powers of terms in arithmetical 

 progression, with a constant co-efficient, viz. '^d. (^ — 1)^/^ 

 (^— 2)c/. &c. (5_„Z:i)^. viz. 



THE POWERS OF NUMBERS IN ARITHMETICA t 

 PROGRESSION. 



or =f^r=py + ^ilpr + Cd'2}T~^^&c. 

 pr^JVd}T = q--+'^dqr +Cd'qT %&c. 

 qr =j:^r ^rr + '^drr +Cd'rr~+8ic. 



^ m m 7)1 J jji Q 



rr=7+l>y=sr + ';ds~ +Cd'sr " + &C. 



» . m m m i m n 



Sr = t + d>7-tr^.'^dtr +Cd'tr~+8CC. 

 m 



tr=8cc. 



FIRST DIFFERENCES. SECOND DIFFERENCES. 



'ydpy '+Cd'py--\scc. 



^d.{?-l)dqy-\&c. 



'^d. {';-!) drr~\ Sec. 

 ^drT +Cd'rr %. &c. 



^d.{'^—l)dsy-^+8cc. 

 ^dsy \Cd'sT-^+&.c. 

 '>^' ^'1' '^d.['^~i)dty +&c. 



•^dty +Cd'tr %&c. 



-,.:,.„ ,., ,>.,.-u4 :.ui.q:iiwm Demonstration. 



