S4 Application of the Fluxional Ratio, ^c. 



In trigonometry, the fourth term may be had by multiplying the 

 third term by the ratio ; in fluxions, the fluent, which is here the 

 fourth term, is likewise had by multiplying the fluxion, which is the 

 third term, by the fluxional ratio. Therefore, since the results are 

 obtained by the same means, the same relation obtains in both, which 

 is that of proportion. 



On fluxions of the higher orders. 



When in the generation of a variable quantity, its fluxion is differ- 

 ent at different points in its production, it may be considered as a 

 fluent, and its fluxion taken, which is called the second fluxion. 

 And when the second fluxion varies, the fluxion of this fluxion may 

 be taken ; and in general a variable quantity admits of as many or- 

 ders of fluxions, as the exponent of the power contains units. 

 Of the second power x^ 

 the first fluxion is 2xx' 

 the second fluxion, 2x'^ 

 the third fluxion, 



Of the third power x^ Of the fourth power x* 



the first fluxion is 3x'^x' the first fluxion is 4x^x- 



the second fluxion, 6xx'^ the second fluxion, ]2x"x'^ 



the third fluxion, 6x-^ the third fluxion, 24^0;-=' 



the fourth fluxion, the fourth fluxion, 24x"^ 



the fifth fluxion, 



In passing from any order of fluxions to the next higher order, in- 

 asmuch as the quantity x' becomes invariable, the exponent of the 

 variable part is diminished by 1 ; hence the ratio for second fluxions 



. (w — l)a;' (n — 2)x' 



is ^ , for third fluxions it is — , for fourth fluxions it is 



{n—3)x- 



, and so on ; generally, 



nx- 

 (1) a;"X — =?2ir"-'a?-, the first fluxion. 



(n — l)x' 

 no?" 'a;'X — = n(n — l)x''-^x'^, the second fluxion. 



(n_2)x- 

 w(«— l)x'-^2;-2X — =w(n— l)(n--2>"-3x-% the third flux- 

 ion. 



