154 



th 

 In general, if the highest members of the n —1 differences 

 III 

 of the y powers of the terms of the arithmetical series of 



Binomials be (as has been shewn in the first, second, and third 



differences) =^f/. ("— l)f/. &c. {^—n—2)d multiplied by the 



powers of numbers in arithmetical progression whose index is 



th 



" — ti — ], then, the highest members of the vj differences will 

 be equal to the common co-efficient multiplied by the highest 



members of the differences of " — ji — 1 powers of terms in 



arithmetical series, = yd. (y—l)d. {r—2)d. &c. (^— jI^K 



ih 



(" — w — l)d. multiplied by the " — 71 powers of numbers in 



arithmetical progression. 



Thus in the Examples the quantities 



m I 



^dt^ ' 



m Q 



^d.{'^—l)dtr ^' 

 ^d.{';~l)d.{';—2)dtr ^ 



~d.{'^~ i)d..8cc.{';—n — \)dt^~^ are found in the series of 



th 

 the first members of the first, second, third, and n orders of 



th 

 differences of the ^ powers of the terms in the arithmetical 



series, in which t is a term, and d the common difference. 



CoK. 



