304 Application of the Fluxional Ratio, 8fc. 



being produced by lines moving with an accelerated velocity. Again 

 let the lines EF, FG and MN, NO, become invariable at the 

 moment when they arrive at the situations AB, BD, and HI, IL. 

 Proceeding with the same uniform velocity, which they before had, 

 they will then generate the quantities ABba, BDdc, and HKA,.ILZr, 

 whose increase is uniform, because equal areas are generated in 

 equal times. The quantities ABba+BDdc, and Hlih+lLlr are the 

 fluxions corresponding to ABDC, and HILK. From the manner 

 in which these quantities are produced, it results that, 



ABba+BDdc : ABDC : ; mih-{-lUr : HILK. 



We have now obtained the proportion between quantities generated 

 by an accelerated velocity, and quantities generated by a uniform ve- 

 locity, disclosing the relation between fluxions and their fluents. 



This illustration may be rendered more general by assuming a prin- 

 ciple, which bears a very near resemblance to motion. It is this. 

 A variable quantity of any kind may be generated by assuming, suc- 

 cessively, and in a regular gradation, every possible value from to 

 x". This will embrace the whole range of fluxional quantities. Let 

 AB = AC be represented by Bx and HI = HK be represented by 



Ux, then, 



2B^xx' : B^a?^ : :2D^xx- : D^x-. 



If we take cubes, the proportion will be, 



SB^a^^a;- : B^x^ ; :SI)^x'-x- : D^a'^. 

 and generally, 



nB"-x"~^x- t B^a;": :wD"a;"~'a;" : D"a;". 



nB"a;"" ' x' iiD^x"' ^x- nx' i r ^ n • 



hence — Ti,nr~ = — T^ii — = — the general formula of the fluxion- 



Bx D'x X ° 



nx' 

 al ratio; hence D^a;" X — =wD"a;''~'a;- is the general equation, by 



which the fluxion is derived from the fluent. 



In this ratio, n represents the exponent of the power, and is ob- 

 tained by adding 1 to the exponent of the variable part of the flux- 

 ional expression ; x represents the root of the given power, and x' 

 represents the fluxion of that root. The existence of this ratio re- 

 sults from the peculiar nature of algebra. The fluxion nB''a;''~' a;- 



nx' 

 consists of the fluent B"a;" combined with the ratio -— • In geome- 



try, these two elements are blended together, and are expressed by 

 the parallelogram GacC. (Fig. 3.) But in algebra the ratio is pre- 

 served distinct during the process, and therefore, since nB''x"'''^x- is 



