185 



orders of the factors being less in those terms than n, therefore 



th -. th 



the n differences of the contents will be the n differences of 

 their highest members, and therefore equal to 



1.2.3.4 n 



1.2.3. . o- X 1.2.3. ./«x 1.2.3. .J X &c. s * 



PROB. V. 



Also, if any series have m number of orders, let d be the 

 last difference, the ii powers of the quantities of that series 

 will have as many orders as ?i x m, and the constant differences 



1.2.3...(?n« — \).mn 



1.2.3.. .»n>" 



X d. For here g + h+i + 8cc.=m x n, and 



n 



a xb,xc.x8cc.=dxdx8cc.=d 



EXAMPLE. 



If of the roots, the differences of the second order be all 

 equal to R, the cubes have their sixth order constant and= 



• — \}' XR=:90R. And this would follow from Problem 

 1 -2 th 



the first, for in the sixth differences of the " powers, we have 



1.2.3.4 5.6 3 , m_3 



the following term, viz. -^, R x ^.(^— 1).(^— 2)P^ and 



(if " = 3, and third and successive differences of the root=0) 



= 90R' 



If 



