114, 



\he first quantity of the series of assumed formulae, Q is the 

 first of the diiferences of the first order, R, the first of the 

 second order of diflerences of the assumed quantities, &c. 

 Therefore, the law of the assumed formulae is the same with 

 that of the series whose orders began with Q, R, S, &c. 

 Therefore all the formulae in the series are rightly expressed 

 as well as P. 



Note. When the given series has no constant order of 

 differences, the expression for preceding quantities, viz. 

 P + Q + R + S + &C. will be an infinite series, I have therefore 

 given the above proof, by shewing that the differences of the 

 assumed, and those of the given quantities, are identical, 

 since the proof which is usually given by a summation be- 

 ginning from the last order of differences, can only be applied 

 where there is such a constant order of diflerences to be found. 



PROP. V. 



If P is one of the quantities in a scries admitting several 

 orders of differences, the first of the differences in the first, 

 second, third, &c. orders of differences of those quantities 

 being Q, R, S, &c. putting (p — d) + (d — eJ+Scc.=p, and 

 (p~d)+2(d— e J +3(e—f)+&:c. or p + d + e+f+&.c. = n. In 



th tk 



all the n differences of the " powers of those quantities in the 



Lemma, the co-efficient of Pr~''xQ^~ xR^'xS x&c. 



will 



