179 



numbers, and prefixing the constant co-efficients of the terms 



th ti' 



in the " powers of a multinomial, we shall have the ^ powers 



th 

 of the successive quantities in the Lemma; in the ?i difl^r- 

 ences of these, all the terms are exterminated, in which, such 

 powers of the triangular numbers are multiplied together, that 

 the number of orders which the triangular numbers admit, 

 being multiplied by the indices of the powers, and the pro- 

 ducts being added together, the aggregate is less than n, viz. 

 when the indices p—d, d—e, e— /, &c. are such that (p— d) 

 + 2{d — e) + 3(e-^'j + &c. is less than n, the co-efficients of 



, ^ f ''' 



Qj' X R X S X &c. will in the ?» differences be extermi- 

 nated ; but when that sum is equal to n, the co-efficients of 

 those quantities will be constant, and are known from Prop 5. 

 Hence, all that can be constant in the first differences will 

 be '^P7~ Q; in the second differences, ^P^ R+Q^.(^-1)Pf ^; 

 the constant members found in the third differences will be, 

 ^Pf-'s + ^,QR?.(?-1)P?-%^3QV.(^-1).(^-2)P?-' 

 the constant members found in the fourth differences will be, 



,>"— 1^ 4.3.2.1 



"(" 1 N^ 1 o "i^^ f ™ 2 4 <! 9 1 ^ , ^^W— 3 



1X1.2.3 f-.(-_l)p^ +ig±iQR?.(^-l).(^2)P^ 



?P? T 



■ +?.(^-l).(?~2).(?-3)P^-*Q 



land so on, for the higher orders of differences. 



TOL. XI. 2 B PKOB. 



