Application of the Fluxional Ratio, fyc. 311 



Since a fluxion always implies a fluent from whence it is derived, 



the only way of obtaining a fluent from its fluxion is by an opposite 



process ; this is called the inverse method of fluxions. Hence when 



the fluxion is derived from any power o( a variable quantity, its fluent 



x nx*. 



is found by multiplying the fluxion by — ; the reciprocal of 



i 



X 



The general equation is nD n x n - , x*X — # =D B x n . 



I. To find the fluent of 27x 2 e x-, we first find the value of n in 



the ratio — ;, and because x 3 6 z* is expressed in the general formula by 



x x 

 z n ~ l z\ therefore 26 + 1=27=71, and — —t^t J hence nD"x n -"V 



9 ' nz* 27x* 7 



x x 



X — =-27x 26 x- X^r =x 27 the fluent required. 

 rtz' 27x* * 



II. To find the fluent of 5(x 3 +x)*X(3x 2 x-+x-); here 4 + 1=5 

 ^^^T^^y henCe ^^-x-X^ = 5(x3+x)^X 



x 3 -J-x 



( Sz2x '+ x ') x H&^r^r( x3 + x y- 



dx* — 2xx X 



III. To find the fluent of -=%( a x-x*) *x(ax'- 



x ax—x 3 



2xx-); here -£ + !=:£ = «, and ~=^ ax . _ 2xx .y hence 



ax - x 



%(ax* — 2xx') 



x i 



nD n x n ~*x*x — m —\{& x — x *) %{ax- — 2xx*) X 



(ax— # 3 ) , the fluent required. 



ayy - s _3 



IV. To find the fluent of — — 3=(a 2 — f) 2 ayy, here 



(a 2 -*/ 2 ) 2 . 



m x a 2 — y* % _. x 



£+1 = — £=n, and — = — 5—— - — ; hence nD n ^ n " , ^*X— - 



21 * ' nx* — £x — 2yy ' nx 



-3. a 2 — V 2 i a 

 (a 3 -*/ 2 ) 2 ayyX =a(a* -j, 2 ) *= t, the 



-£x~2yy- (a 2 -y 3 r 



fluent required. 



V. Thefluentof(2-n)^ , - n x-is(2-n)j: | - w a:-X ratio /^Z^w: 

 or 2 -". 



