Application of the Fluxional Ratio, fyc, 309 



II. To find the fluxion of (x 2 +x 3 ) 4 ; in this example D n x n 



nz* 4(2xx # 4-3x 2 x ,x . nz* 



(x 3 +x 3 ) 4 ,and~ =- — 2 . , ; ; hence D n x n X — =(x 2 +* 3 ) 4 



> J ? z x 2 +x 3 7 X v J 



X ~^+^T— =4(x 2 +x 3 ) 3 X(2xx'+3x 2 x-). 



l 1. 



III. To find the fluxion of (ax— xx) 2 ; here D n x n =(ax — xx) , and 



£( 



x ax — xx 



ax # — 2xx* 



— h hence DVx— =(ax~xx)^X — 



ax — xx 



2(ax-xx)^ 



IV. To find the fluxion of — ?=a(a 2 -y 2 )~^; here DV 



a(« 2 -y») and— : = a _ 3 ,henceDVx— =a(tt 2 -y 3 ) 2 X 

 - J X - 2w ayy 



«— .r („'-»>)* 



a -A -£x — 5y 4 y 

 The fluxion of r is a(a 3 — y 5 ) 2 X ratio- 



(a*-y*f 



3 *|5 



y 



5ay*y 



2(a 3 — y 5 ) 



3 

 2 



1 1 Xx" £* 



The fluxion of - is z" 1 X ratio — = — — • 



X XX 2 



1 1 3x* 3x 



The fluxion of — r is — rr X ratio 



ax 3 ax" 3 % ax 4 



The fluxion of %(ax — a 2 )* is §(ax - a 2 )^ X ratio 



lax- 



ax— a 



a(az — a 2 yz\ 



1 1 



The fluxion of — j- r (x m + a 2 ) n+l is — rr(x m +a 3 ) n+l X ratio 

 » ~T— 2 =(z m +a*) n xmz m -'z: 



X Ttt^ % * 



■ 



m . ^ (o n, 4-« m ) B+, . (a m +z m ) n+1 (n+l)mz m - , z- 



The flux.on of w(n+1) is " w(n+1) X ratio ^^ - 



(o m +2 m ) , X2 m "'r. 

 When the fluent is the product of two or more flowing quantities, 

 as xyz, the fluxion of each is to be taken seriatim, the other remain- 



Vol. XXIV.— No. 2. 40 



