166 



powers of the first members of the Roots, will be the n powers 

 of the next lesser terms of the series, viz. 



FIRST DIFFERENCES. 



p =q + d^ =q +7idq +Cdq + &c, ^_j ,„_2 



ndq + Cdq + &C. 



q =:r + d' =7- +ndr +Ldr +&.C. _ , _^ 



ndi" + Cdr + &G, 



n— 1 ^ ? n— 2 



r =s+dl =s +«ds +Cc?s +&C. _ ^ _ 



«£?s + Cf/s + &c. 



5 ZZ&C. 



Hence, if we take away from the n powers of the Bino- 

 mials the first members of those powers, we sha^I have taken 



th 

 from them the n po%vers of the next lesser terms of the arith- 

 metical series, and the remainders are the first differences of 



th th 



the ?j powers of the Binomes. Therefore the ;« — 1 differences 



th th 



of the remainders are,the n differences of the n powers, but' 

 those remainders involve only lesser powers of the first mem- 

 bers of the Binomes, which first members are in the same 



arithmetical series, and tlie highest of those powers are the 



th 

 n — 1 powers of the terms of that arithmetical series, the co- 

 efficient of these powers being nxJ. 



Therefore in the case where the Index n is equal to 2, the 

 second differences of the squares will be common, for they 

 are the first differences of the terms of an arithmetical series 



multiplied 



