310 Application of the Fluxional Ratio, &fc, 



ing quantities being considered as a coefficient ; then the several 

 quantities thus arising will be the fluxion sought. 



Ex. to find the fluxion of xyz. Taking x, the ratio is — = -» 



and xX— — 2;", which is the fluxion of x; hence yzx' is one part of 



nx' y y 



the fluxion. Taking y, the ratio is — =75 and yX — =y',ihe flux- 

 ion of y ; hence xzy is another part of the fluxion. Taking z, the 



nx' z' z' ^ 



ratio is — =■—> and zX — ^^z-, the fluxion of z; hence xyz' is 

 X z z ' > u 



another part. Therefore, yzx' -\-xzy' -\-xyz' is the whole fluxion of 



xyz. 



Ex. to find the fluxion of x^y^. The quantity x^Xthe ratio 



3a;* . . . by 



— =^x^x', the fluxion of x'^ ; and the quantity y^ X the ratio — = 



X " y 



5y^y, the fluxion of y^. Hence 3y^x^x'-\-5x^y'^y is the fluxion 

 of x^y^. 



X 



To find the fluxion of the fraction -=xy"^. Taking x, the ratio 



. nx' X- X' , n . r 1 ^' • 



is — =— > and a;X-=a;", the fluxion 01 x: hence y 'x'=— is one 

 XX x ' y 



nx' y 

 part of the fluxion. Taking ?/"', the ratio is — =— — , and y~^ X 



X y 



y y . xy . 



— — =. —y 2y. — _— -, the fluxion of w ' : hence — ^r is the other 



y ■ y i) y. U y y2 



X" xy yx' — xy . 

 part of the fluxion; therefore — —--= — is the fluxion of the 



f ' y y' y^ 



fraction -. 



y 



To find the fluxion of the fraction ^=^x^y~^. Takingx^the 



nx' 2x" 2x' 



ratio is — — — , and x" X — =2xx' the fluxion of x^ ; hence Sv^cca;* 



XX X 7 J 



nx' 2y 

 is one part of the fluxion. Taking y~^ the ratio is — = — — , and 



X y 



^"^'X — — =—2y ^y the fluxion o(y'^, hence —3x"y *?/• is the 



2xx' 

 other part of the fluxion. Therefore 2y~^xx' — Sx-y~'^ys= — — — 



2x^y 2yxx' — ox-y' . , „ . „ , ^ . 3;^ 

 — 7~= 2 IS the fluxion 01 the Iraction — . 



yA yA y3 



