VOL. LXXXV.] PHILOSOPHICAL TRANSACTIONS. 579 



... , ,, 71 + I 71 + \ 71 + \ —)■ 71 + 1 n + 1 — r.n + 1 — -Jr 



right hand, equal to 1 ; -^^^ ; —^— X ~ ; -^^— X — X -^ ; 



&C. respectively : and this will be fully demonstrated when we have proved that 

 all the terms of products in any perpendicular line, in which the exponent of r 

 in the denominators is t, being multipled by , are equal to the whole 



of the next perpendicular line of products towards the right hand. 



To do this in a manner applicable to any part of the series concerned, and to 

 avoid numeral coefficients, which would obscure and encumber the general rea- 

 soning, it is necessary to find the value of the numerator of "- — - — '- in terms 

 of A, B, c, D, &c. and of a, b, c, d, &c. and to ascertain the relative values of 

 a, (3, y, i, &c. and that we may do this with due precision and perspicuity, it is 

 proper to fix on 2 contiguous perpendicular lines of products. By reasoning 

 from which, Mr. R. effects this in the next or 17th article. 



18. The relative values therefore of «, j3, y, &c. next claim our attention ; 

 and from the nature of the series it is 





a. 



D — 5 ' V — 6 



Also 1 = «, - = «, J = p, - = y, 5 = J, g = £, ^ = «;. 



The only thing necessary now, is to reduce the denominators of the first side, to 

 the denominators of the 2d, and in such a manner as to make the parts on the 

 first side, which have the same numerators, unite : this being done, and the parts 

 all arranged in order, as in this article, it is at length obtained that 



G + aF OF + eB j^ 6e + CD j^ CD + rfc j_ dc + eB , eBjf/A , J.\ + g 



G j^ QF j^ 6f. , CD , rfc , CB , /a , g 



1 9. This being proved from the relations between the 2 contiguous perpen- 

 dicular lines, and these relations being the same between any 2 perpendicular 

 lines whatever (for they are as regular and certain as the laws of continuation in 

 the multiplicand and multiplier with which we set out in the 13th article) it fol- 



n -4" I ^ mr 



lows that if ^=^ exoress the whole of any perpendicular line, the next 



n+ I - (m + l)r 



perpendicular line to the right will be tJLSl ^^^ ^ i)rm-^-i " ^^^ 



1 ".-1 --I 1-z 



therefore the series x" ■■\--zx'' +- X ^ zV + &c. being multiplied 



by the series x -\- - zx' + - X '-^ zV + &c. the product will be 



1 r ,1 



V ■ 



2 



4 E 2 



