310 Application of the Fluocional Ratio, SfC. 



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 — = -» 



X X 



X* 



and xX-=r, which is the fluxion of x; hence yzx m is one part of 



X 



nx* y y* 



the fluxion. Taking y, the ratio is — =-> and yX7-=y*,the flux- 



^ y y 



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



fix* z m z* 



ratio is — =-> and zX~ '=z\ the fluxion of z; hence xyz* is 



x z z ' * * 



another part. Therefore, y^+x^y'+fcyz' is the whole fluxion of 

 xyz. 



Ex. to find the fluxion of x 3 y 5 . The quantity x 3 X the ratio 



by 



=3x 2 x', the fluxion of x 3 ; and the quantity y 5 X the ratio — = 



5y 4 y, the fluxion of y 5 . Hence 3y 5 x 2 x* + 5x 3 y 4 y is the fluxion 

 of x 3 y 5 . 



x 



To find the fluxion of the fraction -=xy~\ Taking x, the ratio 



nx m x' x* b x* 



is — =—? and xX-=r, the fluxion of x; hence y -l ar=s— is one 

 xx x 7 y 



fix* V* 



part of the fluxion. Taking y~ ! , the ratio is — = — — , and y~ l X 



x y 



y* . y xy 



— T" = — y 2 y ' = — — j the fluxion of y~ l ; hence — — is the other 

 u j if 



X* XXI* UX m - XII* 



part of the fluxion ; therefore — — - - —- r-^- is the fluxion of the 



3x 



y r V 



fraction -. 



y 



x 2 



To find the fluxion of the fraction — = x 2 y~ 3 . Takingx 2 the 



y 



nx' 2x- 2x # 



ratio is — = — , and x 2 x — =2xx # the fluxion of x 2 ; hence 2y 3 xx- 



X X . 



7VT m ^77" 



is one part of the fluxion. Taking y~ 3 the ratio is — = — — , and 



x. y 



3y 

 y 3 X — — = — 3y 4 y- the fluxion of y 3 , hence — 3x 2 y~ 4 y is the 



2xx # 

 other part of the fluxion. Therefore 2y~ 3 xx* — 3x 3 y-*y 



3**r 2yxx--3x 2 y , x* 



•■ . = ~i is the fluxion ol the fraction — ♦ 



y y y 3 



