]80 



PKOB. II. 



If in a series of Quotients one of which is ^ the dividends 



admit several orders of differences, of which the differences 

 corresponding to p in the several orders, are g, r, s, t^ Sec. rf„,&c. 

 and in like manner, if of the differences of the divisors, those 

 which in the several orders correspond to P, are Q, R, S, T, 



th 

 &c. D^, &c. Then, all the constant parts in the ji differences 



of the terms of the series of quotients are, 



, 1.2.3.. .M -■ _ _i— 1 



1.2.3...n >-X— IXP 



lXl.2.3...(7t-l)?D„_, 



+&C. 



1_^2 . 3 . . . 71 

 Ix 



+ &C 



^-lxl.2.3...(n-l)PSS: |x-lX(-l-l)p-'-%&c. 



tk 



For, if the — 1 power of P+Q+R+S+&:c. (which is of the 

 same form as above, only substituting — 1 for ") be multiplied 

 by p-\-q+r+s + Scc, and the several multiplications of the 

 terms be performed, according to the powers of P, taking 

 also into account the places of the members of the multino- 

 mial multiplier. In the place of the quantities included in 

 the several lines in the proof of the preceding Problem, we 

 shall now have the quantities, in the following expression, 



contained 



