1G8 

 the remainders, or the ?* differences of the given powers, will 



Ih th 



be the «— 1 differences of n times the n — \ powers, wdich 



ih 



are the highest members in those remainders: for the n — 1 



differences of the members containing lower powers, will be 



the differences of common differences, and therefore =0. If 



then (as we have shewn in the second and third powers) the 



th th 



n — 1 differences of the n —1 powers be =lrf.2(i...«- -Irf, the 



th th 



n differences of the n powers =lc/.2(i...n — \dxnd or =: 



1.2.3.,.?* — i.nx-d. 



PROP. II. 



If there are n number of arithmetical series, whose com- 

 mon differences are respectively a, b, c, d, &c, let all the 



difi'erent corresponding terms of the several series be mul- 



th 

 tiplied together, the n differences of the products will be 



constant and equal to 1.2. 3. ..nxaicc^ &c. 



For, let A, B, C, D, &c. be the correspondent terms of 



the different series, the products, will then be 



3rt + A X 3^ + li X 3c + C X 3</ + D X &c. 



2« + A X 2/^+ 13 X 2c + C X 2(/+ U x &c. 



a + Ax 6 + Bx c + Cx d + Dx&c. 

 Ax Bx Cx Dx &c. 



And 



