172 



will now be the contents of as many arithmetical series as 



th 

 gq + hr + ks + &.C. which sum putting equal to 7i, the n differ- 

 ences of the products of the corresponding terms of n arith- 

 metical series, which (by the second Prop.) = 1.2.3. .7/, will be 



th 

 the n differences of the products of the powers of the contents. 



But, (from Prop. 3.) if we divide those products of the powers 



of the contents by 1.2.3.. .g-'^x l.ii.3.../tVx i.2.3...A;'^+&c. we 

 shall have the products of the powers of the corresponding 

 terms of the different series of triangular numbers, whose 



orders are g, h, k, &c. Therefore the ii differences of these 



will be 1.2.3...7?, divided by 1.2.3...g'^'x 1.2.3.../A'"x 1.2.3...Al'x 

 1.2.3.4 n 



■'" 1.2.3.. .o"^'xi.2.3.../i^^xi.2.3...A?'x&c. 



LEMMA. 



The quantities in any series can be expressed in a multi- 

 nomial form in terms of triangular numbers and of the first 

 of the several orders of differences of the quantities : viz. let 

 P be the first quantity taken in a series, and let Q, R, S, 

 &c. be the first of the several differences whose orders are one, 

 two, three, &c. ; the following general formulae will express 

 the preceding and subsequent quantities in the series, viz. 



GENERAL 



