173 



GENEUAL FORMULA. 



Ih 



V + nQ4. "Cn+v 'R+ n.('t-^\).Cn+v S &c.=» prcccding quantity 



+&C. + &c. 



+ &c. + r»+i)C''+2) D+&c. 



1.2 n 



+ &C. +(n+l)D +&C. 



n 



+&C, + D +&C. 



n 



P = the first of the given quantities. 



P— Q 



p_2Q + R 



p_3Q + 3R — S 



&c.— &C.4- &c.— &c. +&C. 



p ,)Q4.« Y"— U R n.Cn—\).Cn—2) ^ &c. = « Succeeding quantity, 



^ 1.2 1.2.3 



Por, since in those formulae, the co-efficients of Q have their 

 differences equal to unit, and the co-efficients of R, their se- 

 cond differences, =1, &c. if for P, Q, R, &c. in the formulae, 

 we substitute Q, R, S, &c. respectively, we shall have the 

 first differences of the quantities ; and by substituting R, S,T, 

 &c. we have the second differences, &c. Therefore, P being 



2 A 2 the 



