94 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 n flucions of the higher orders. | 
oan 3 in ‘the generation af a variable quantity, its fusion i 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. 
the second power =x? 
the first fuxion is 2zra- 
the second fluxion, 2x°*. 
; the third fluxion, = 0 
Of the third power a2* — Of the fourth power at 
the first fluxionis 3x?z- the first fluxionis 42x%2° 
the second fluxion, 6x2*? the second fluxion, 12x?2*? 
the third fluxion, 6x"? the third fluxion, 24aa-* 
the fourth fuxion, 0 the fourth fluxion, 24a! 
the fifth fluxion, 0 
In passing from any order of fluxions to the next higher order, in- 
asmuch as the quantity 2° becomes invariable, the exponent of the 
variable part is diminished by 1 ; hence the ratio for second fluxions | 
m—1)x° - n—2)zx° 
is ar, for third fluxions it is! . de 
(n—3).xr° 
, for fourth fluxions it is 
' and so on ; generally, 
(1) ex ~=na"-'2", the first fluxion. 
n—1)x° ine 
( ) =n(n—1)z"~?.2*?, the second fluxion. 
= 2)a* 
L 
nx" ' a x 
n(n—1)2"~2 459 x =n(n—1)(n—2)z"-*2"*, the third flux- 
ion. 
