J69 



And those factors being multiplied, tlie products will be 

 S" X ahcd Sec. -\-3"~^ x Abe &c. + aBc &c. + &C.+ABCD &c. 

 2" X abed Sec. + 2''~' x AZ*c&c. +aBc&c. + &C.+ABCD &c.^ 

 l"xaZ>cd&c. +l"~ x'AbcSic. +aBT&cr+&c.+ABCD &c, 



+ABCD&C. 



The highest terms in those products are the n powers of 

 the natural numbers with the constant co-efficient abed &c. 

 which is the product of all the common differences. The 

 subsequent terras are the inferior powers of the natural num- 



bers, with certam constant co-efficients. But in the 71 differ- 

 ences, all the inferior powers are exterminated, the terms 

 which are the differences of common differences beint^ =0. 



Therefore, the 7j differences of the contents are the 71 differ- 



ences of their highest members, or the n differences of the 



th 



n powers of the natural numbers with the constant co- 

 efBcient, abed 8cc. or=:1.2.3...nx abed &c. 



PROP. III. 



> 



The series 1, ^H, f n+p.m+Q j^ m+D-rn+TCn-^-sj ^ Scc. will have 

 the n differences of its terms common and equal to unit, 



(which is the definition of triangular numbers of the n order) : 



For 



