Application of the Fluxional Ratio, ^c. 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 of a variable quantity, its fluent 



X . nx\ 



is found by multiplying the fluxion by -^ the reciprocal of — 



The general equation is nD"?c" 'a;* X~=D"x''. 



I. To find the fluent of 2'7x'^^x-, we first find the value of n in 



the ratio —77, and because a; ^ ^x- is expressed in the general formula by 



X ^ 

 x"~'x-5 therefore 26 + l=27=w, and — :=^^;77. ; hence wD"x" 'x' 



X — =^27x^ ^^' X^^^iT- =x- ■' the fluent required. 



nx' 2lx' ^ 



II. To find the fluent of 5(x3+a;)-'x(3.r3a;-+z-); here 4+1=: 5= 



X X3-|_X X 



*^'^"^m- = 5(3x^^M^)' ^"""" «D"x"-'x-X- = 5(x^+x)^X 



(3x^x.+x-)X^^^) = (-^+x)^ 



ax' —2xx' _! 



III. To find the fluent of -=^{ax-x^) -x{ax'- 



2{ax — x^Y 



2XX-); here -i + 1=^ = 71, and —^^i^J._2xx-y ^^"^^ 



X 1 ax-x^ 



n-D'^x-^x-x-=i{ax-x^) ^ X(ax. - 2:ra,-) X ^^^^7=2^) = 



(^ax — x^y, the fluent required. 



ayy 3 



IV. To find the fluent of ^=(«^— 2/^) ^«2/2/*; here 



— 1+1 = — ^=«, and — = — 7-— -7^ — ; hence nD"a;"-»a:-x — 



_3 a^—y"^ ^_i a 

 =(a2_j/2j 2^^j^.x ^ =a[a^ -y^) ^= i' ^^e 



— ^X—2yy («^-2/^) 



fluent required. 



a; 



V. Thefluentof(2-w).r'-\r-is(2-w):r'-"j^-X ratio ,^ _ . 



