153 



bers in arithmetical progression, whose Index is ^ — 1, and 



since the highest members of those " — 1 powers are (" — 1) d 

 midtiphed by powers whose index is " — 2. Therefore the 

 highest members of the second differences will be the common 

 multiplier "d. (^ — l)d multiplied by powers whose Index is 

 " — 3. Again the highest members of the differences of the 

 ^ — 2 powers of numbers in arithmetical progression are 

 (y — 2)d. multiplied by the powers whose Index is ("' — 3), 

 Therefore the highest members of the third differences of the 



^ powers =^J(^—l)i.(^—2)rf. multiplied by the ^—3 powers 

 of numbers in arithmetical series. 



GENERAL EXAMPLE. 



th 



The n — 1 differences. 



^d . {^—l)d . . &c. {j—n-2)ds7~"^' + &c. 



^d . (?— l)d . . &c. (^— n- -2) dt r-"-' + &c. 

 &c. 



th 



th 



The n differences or the differences of the n— 1 differences, 

 ?d, (^-^1) d. . &c. (^— »=^) d. C^—^T^) dt^~\ &c. 



In 



