182 



pressing the powers of the triangular numbers which are the, 

 variable co-efticients of d^, Q, R, &c. as in Prop. 5, the fol-. 

 lowing quantity will be found, viz. 

 ».(»— l).(?i— 2) .. 3.2.1 _l.,_2.— 3...— /. 



■"■•"" > \ ~} ' /. "■■■ ~ Y, ' — —"•-■' ' -■ I. ■ . ■ 



1.2..vxT''~xT^ ~x^^'l<^c. i.2.3...p—dxl.2 3...f/— ex&c. 



X P ^ xd X Cf~ X R ~^ X &c. will be constant in all the 



th 

 n differences of the quotes. Thus, the several quantities of 



th 

 this kind which will be constant in all the 4 differences of the 



terms of the series of quotients, are viz. 



PROB. III. 



Let there be any series of products, the successive factors 

 admitting several orders of differences, viz. Let one of the 

 products beAxBxCxDx &c. and in the series of factors to 

 which A belongs, let the differences corresponding to A, in 

 the several orders, be Oj, a^, a^, &c. o,,, &c. And of the 

 series of factors to which B belongs, let the differences cor- 

 responding to B, in the several orders, be 6,, b„, b^, &:c. b^, &c. 

 And in like manner, also, of the series of factors to wLich 

 C belongs, let the differences corresponding in the several 



orders 



