98 Application of the Fluxional Ratio, &c. 
If the foregoing series marked (2) (3) (4) (5) be compared with 
the orders of fluxions marked (1), a remarkable coincidence will be 
observed. If the nature of the relation, which exists between flux- 
ions and their fluents, is sought for; if it should be asked, what is 
the rationale of the result ? and why does this coincidence take place ? 
the answer will be, that these series contain the elements of the ra- 
tio — by the multiplication of which, or its modification, into the 
expression of any order immediately preceding, the fluxion of the or- 
der next following is produced. 
To illustrate this, let them be brought into one form, ani exem- 
plified in the function x". When properly arranged they will stand 
thus, 
Orders of ects : 
quire § fluent x"\nz"~'x*\n(n—1)2"~?.2°? n(n — ¥)(n—2)a"- 2x** 
: ‘ . oe r a3 
Torn ¢ ena" 2"\n(n— I)a"2 5 > |n(n—1)(n—- 2)a"* 53 
Bi 1 = —L)(n—2 
a i alae en ( ae ney a Aes ) x" 43, 
Here in the series of ae y is aunt by 2”, aa h by z*, 
and z* by 1; in the binomial series A is represented by 2°.’ 
In the cndern of fluxions, the fluent 2" multiplied by the fluxional 
ae 
ratio a produces nz"~'z: the first fluxion ; this, considering %* a 
ot + ae y PEE Sv: & 5 a i 1) 
. = 
i wf 
» the ratio for fluxions of the 
second order, produces tae —1)z"~*z** the second fluxion. Molti- 
(n ee 
third i we obtain afiine’s aya “je x, the third fluxion, and 
sO on. 
Ly]. : Hi is 
In the series of Taylor "> is expressed by nz"~', and by 
n(n —1)x"~*, and By n(n—1)(n—2)z"-%, &c. In the second 
term there is a complete coincidence. In the third and fourth terms - 
plying the second fluxion by , the ratio for fluxions of the 
a = 23 ‘A 
if the divisors of the factors 3 as in the series of Taylor be with- 
drawn, we have‘the second and third fluxions; if they be transfer- 
red to the factors in the same series expressing the coefficients of 
