184 



Note. Besides the constant quantities expressed in tlie 

 above several formulae, the actual differences will consist also 

 of certain variable quantities, except in the following cases, 

 in which the actual differences themselves will be constant. 



PROB. IV. 



Let the greatest number of orders which a series will admit 

 be g-, and let the constant differences be a. Let the greatest 

 number which another series will admit be h, and constant 

 difference be b^. And let another series admitting i orders 

 have a constant difference c^. &c. Then if g-{-h+i+ Sec = n, of 

 the contents of the corresponding terms in all those series the 



actual (iifferences of the n order will be constant quantities, 

 _ 1.2.3.4 71 



^'^'^ ^""^-ri^Zi^TilXXrTi^riT&F. xa.xKxc,^ &c. 



For, in the contents of the form (A + &c. H-fl^)x (B + &c. + iJ 

 (Ch-&c. + c)x &c. the product of all the constant differences, 



or a x6. xc.x&c. will be the common co-efficients of the 

 e * ' 



products of the highest orders of the triangular numbers iu 

 the successive expressions deduced from the Lemma, and the 



th 



n differences of the products of those numbers are constant 



th 

 by Prop. IV. But the n differences of the terms containing 



all except the highest orders are exterminated, the sum of the 



orders 



1 



