183 . 



orders be e,, c^, c^, &c. c„, &c. &c. All the constant parts 



tk 

 in the n differences of the contents will be, viz. 



fa ;BCDE Scc. + AiCDE &c.+ABc„DE &c.+&c.) 



+n(a„_^b^CDE &c. + a,6„_,CDE &c. + &c.) 

 ^ ""-'(ffl_^<;.CDE &c. + a^b„_/:DE &c. + &c.) 



+ &c. 



+ n.{n—l) ("o„_3fejC,DE&c. + a,6„_oC,, DE&c. + &c.) + &c. 



+ "'"Txi''r'' ("n-3¥.DE &C. + &C. + &c. 



+ &C. +&C. 



For, instead of the powers in Prop. V. we shall now have 

 a series of products in which the factors are of the forms 

 A + a, + a., + &c. B + b^ + b^ + &c. C + c^ + c^+Scc. and in which 

 products, from the nature of multiplication, the combinati- 

 ons of differences are multiplied into all the other first terms 

 but their own, &c. And, of the triangular numbers, which 

 in the successive contents of this form, are the co-efficients 

 of those differences, a^, a,,, &c. h^, b„j &c. &c. whenever the 

 combinations are such, that the sum of the orders of differ- 

 ences which the triangular numbers admit is equal to 7i, the 

 /* 

 n differences of the members containing these combinations 



will be constant. And if g+h + i + 8cc. = ii, in all the /i dif- 

 ferences we have the following quantity, viz. 



Note. 



