177 



CHAP. II. 



Problems for finding, per Saltum, all the constant parts in the 

 Differences of any order. 



PROB. I. 



th rt 



All the constant parts in the n differences of the " 

 powers of quantities not increasing uniformly, will be 



»-, + 1.2.3. ..» QD 

 '^ „ Ix l.2.3...(n— 1) n-l 



,. 1.2.3 ...» ^^^ U^.(^_1)P"-^ 

 1.2 X 1.2.3. ..(«-2) n-^y 



+&C. 



1.2 X 1.2.3 . . . (ra— 2) '^~i 



1.2.3 ■■■« oRn J- X ?.(?-] ).(^-2)P"-'+&c. 



1 X i.2 >t-1.2.3...(?J— 3) n-3*' 



+&C. 



which may be also expressed in the following form, by begin- 

 ning with the constant parts in which P has the lowest index, 



'"—n^ 



viz. ^.(^_.])...(«_,i_l)PF Q" 



-^ "•r'^.^"r^ ?.(^-])...(^-;^).p"-^^xQ:-^s-f&c. 



the 



