128 VHILOSOPHICAL TRANSACTIONS. [aNNO 1751. 



last ages received, the method of series may be justly deemed one of the most 

 considerable i since not only the doctrine of chances and annuities, with some 

 other branches of the mathematics, depend almost entirely on it ; but even the 

 business of fluents, of such extensive use, would, without its aid and concur- 

 rence, be quite at a stand in a multitude of cases, as is well known to mathe- 

 maticians. 



It is for this reason, that the celebrated binomial theorem, for converting 

 radical quantities into serieses, is ranked by many among the principal discoveries 

 of its illustrious author ; seeing, by it, a vast number of fluents are found, that 

 would otherwise be impracticable : nor is there any case, however complex, to 

 which it may not be extended. 



It is true, when 2 or more compound radical quantities are involved together, 

 the operation, by having two or more serieses to multiply into each other, be- 

 comes very troublesome and laborious; and, what is worse, the law of continu- 

 ation, by which a part of the labour might be avoided, is exceedinglv hard, if 

 not impossible, this way to be discovered. In the following paper something is 

 attempted towards obviating the said inconveniencies. 



Prob. 1. — To find a series exhibiting the value of 



(1 -h j)"- X (1 -}- 1^)" X (1 -h ^)' X (1-1- ^)', &c. in simple terms ; x being in- 

 determinate, and a, b, c, d, m, n, p, &c. any given numbers, whole or broken, 

 positive or negative. 



PutM=(l4-^-)«,«;=(l-t-^r, y = (H-J),z = (l-H|), &c. 

 Also let A = uwyz, &c. (= the quantity proposed) 



Then, in fluxions, ^ = uwyz &c. -\- uwyz &c. -|- uwyz &c. -}- uwyz &c. &c. 

 Which equation, divided by the preceding one, gives 



A iL + i + -L+1 &c. 

 A u to ' y ' i 



But since u = (1 + -)"*, we have « = wii X (1 -|- -)**"' ; and therefore 

 i =^ X {.+'-)- =V X O-i+i;- 5 + ?_&c.)b,dl,lsio„. Andin 

 the same manner it appears, that ^ = yX 1 — |--|-p&c. &c. 



Hence, our equation, by substituting these values, becomes 



['m mx j^ mx^ mx' . T 

 la a^ * a^ a* 



k . \ 1 nx , nx' nj? 



C^ ' t^ C* J 



&C. &C. &C. &C. 



