193 



the dift'erences of those differences, &c. as often as the index 

 of the highest power denotes, the co-efficients which ai'e con- 

 nected with the inferior powers will be exterminated, and 

 the highest co-efficients alone will be involved in the last dif- 

 ferences, with the constant differences of the powers. For, 



th th 



all the ti differences of the n powers of numbers in arithmetical 



n 



progression are constant and — 1 x 2 x 3 x ^...(n — ]).nX^ 



A general statement of Newton's rules for finding divisors 

 can be easily deduced from the foregoing observations, as 

 follows : 



Substitute successively for x in the proposed, the terms of 

 an arithmetical series, 3, 2, 1, 0, -1, -2, -3, until the number 

 of terras is greater then the index of the divisor required; 

 place the numbers resulting from the substitution with all 

 their divisors, as well affirmative as negative, opposite to the 

 correspondent terms of the substituted series ; take the dif- 

 ferences of those divisors, and the differences of their differ- 

 ences, &c. ; if tlie differences of any series of divisors be com- 

 mon, when the number of orders of differences taken is equal 



4o n or the index of the polynome sought, that difference 



th 

 being divided by the last difference of the n powers of the 



terms of the natural series or by 1.2.3...n, the quotient should 



be a divisor of the highest term of the proposed, and if so 



should be made the common denominator of the co-efficients 



of 



