COMPARISON OF VARIABLE QUANTITIES. 11 



used, but the evanescent increment is called a difference, and de- 

 noted by d or h, and tlie variable quantity is conceived to consist of 

 the entire sum or integral of such differences, and marked /, as xzz. 



J dxy oxjlx. This mark has the advantage of differing in form from 

 the short *, which is used as a literal character. See 229. 



47. Theorem. When the fluxions of two quantities 

 are in a constant ratio, their finite increments are in the 

 same ratio. 



For if it be denied, let the ratios have a finite difference ; then if 

 the time, in which the increments are produced, be continually di- 

 vided, the ratio of the parts may approach nearer to the ratio of the 

 fluxions than any assignable difference, for that ratio is their limit 

 (46), and this is true, by the supposition, in each part; therefore the 

 sums of all the increments will be to cacli other in a ratio nearer to 

 that of the fluxions than the assigned difference (36). 



48. Theorem. The fluxion of the product of two 

 quantities is equal to the sum of the products of the 

 fluxion of each into the other quantity. 



Or {xy)":^yx-\-xy. Let the quantities increase from x and y to 

 x-\-x' and y-\-y, then their product will be first xy and afterwards 

 xy-\-yx! ■\-xy-\-x/y\ of which the difference \^yx!-\-xy-\-x'y ,2iW^ the ratio 

 of the increments of a: and xy is that of a:' to yx' ■\-xy-\r^'y\ or, when 

 the increments vanish, to yx'-^-xy, since in this case xfy vanishes in 

 comparison with xy. But x' \ {yx' -\-xy)'. \x : {yx-\-xy\ and the fluxion 



isrightly determined (46); for since i-= 2-, ^=^(18); but 



:t^ X xf X 



^'=2? (18),andl!£±2'=2±!:2(15). 



XX x' X 



Scholium. It is also obvious, that the fluxion of any quantity xy 

 is equal to the sum of the results obtained by multiplying it by the 

 fluxion of each variable quantity, and dividing it by that quantity : 



\km,{xyy—xy \JL -j-iLj ; {xx)"=zxxy^ Jf.JL^-=2xx. 



