170 



Tor if v,c take as an example of the preceding proposition, 

 the contents, &cx &c x &c..&c. 



4x5x 6...{n + 3) 



3x4x5...(n + 2) 



3x3x4..,(7H-l) 



1 X 2 x 3 . . . ?j 

 in which the different arithmetical progressions are the na- 

 tural series, and also A, B, C, D, or the corresponding 



terms are the natural series 1, 2, 3, &c. to ii. One of 



th 

 the contents — 1.2-3. ../», which is equal to the n differences 



of the contents. Therefore, if we divide the quantities 



].2..rfi, 2.3. ..(n + 1), 3.4...(m + 2), 4.5...(m + 3), &c. by 1.2...7t 



th 

 unity will be equal to the 7i differences of the quoticnt-s 



'l'l.2 1.2.3 



fROP. IV. 



If the g powers of the natural numbers, and the r powers 



of the corresponding triangular numbers which admit two 



th 

 orders of differences, and the corresponding s powers of the 



triangular numbers having three orders, &c. be all multiplied 

 as follows: 1 x I'^x 1 x &:c. 



2''x3'"x4'x&c. 



S'x^'xlO'x&c. 



&C.X&C.X&C.X&C. And 



