98 



.Application of the Fluxional Ratio, &fc. 



If the foregoing series marked (2) (3) (4) (5) be compared with 

 the orders of fluxions marked (1), a remarkable coincidence will be 

 observed. If the nature of the relation, which exists between flux- 

 ions and their fluents, is sought for ; if it should be asked, what is 

 the rationale of the result ? and why does this coincidence take place ? 

 the answer will be, that these series contain the elements of the ra- 



. wa;* ... 



tio — by the multiplication of which, or its modification, into the 



expression of any order immediately preceding, the fluxion of the or- 

 der next following is produced. 



To illustrate this, let them be brought into one form, and exem- 

 plified in the function a?". When properly arranged they will stand 

 thus, 



Here in the series of Taylor, y is represented by a?", and h by z', 



and a;" by 1 ; in the binomial series h is represented by x-. 



In the orders of fluxions, the fluent x" multiplied by the fluxional 



nx' .... 



ratio — produces na;""^ a;' the first fluxion ; this, considering x- a 



{n — l)x 

 constant quantity, multiplied by , the ratio for fluxions of the 



second order, produces n{n — l)x'"~-x-^ the second fluxion. Multi- 



(n-2)x- 

 plying the second fluxion by ] — — , the ratio for fluxions of the 



third order, we obtain w(w— l)(w — 2)a;"~^x*^, the third fluxion, and 



so on. 



M . [yy 



In the series of Taylor -7 is expressed by nx^~^, and ~r2"by 



\yV 



w(w— l)x"-Sand^7^by n{n — l){n-2)x"-^, he. In the second 

 term there is. a complete coincidence. In the third and fourth terms 



X'^ X'^ , . rn 



if the divisors of the factors ^, q:^, in the series.of Taylor be with- 

 drawn, we have the second and third fluxions ; if they be transfer- 

 red to the factors in the same series expressing the coefiicients of 



