FLUXIONS. 401 
- line BC bemg e described with a variable velocity, that ding increments, so that the fact, that every Fundamen” 
is always less than V, and CD with a variable velocity Varishle, quaitity "hes ednicthing related to it, which | 
than V; the line BC will be less may be made the subject of mathematical investiga- pl 
tion, namely, the velocity with which it increases, re- , 
tal 
“Principles. that is always greater 
_ “"Y~" than s, and the line CD greater than s, and conse- 
will be 
8 cD 
s 
greater than —; therefore — >= 5 and — <= 
cD 
ca’ 
of every function whatever 
ing velocity. 1f, however, the velocity decrease, then, 
Sarai, coca: Sree Ss be found that 
BC a4 cD r 
>< te? and — > —7: So that, in every case 
the ratio of the velocities of the generating points 
at any contemporaneous position, is always between 
the ratio of any finite increments in an in- 
terval of time ended when they arrived at that po- 
‘sition, and the ratio of finite increments 
inan interval of time that when they 
is, the ratio of the velocities is greater than the one, 
-bat less than the other of these ratios; provided that 
prover mecemacy eS described, the velocities are 
s increasing, or always ing. 
e have observed that the pe ary 
BC, snl pena weenie cue tre Hooves ena 
because of the acceleration or retardation of the motion; 
the less, however, these increments are taken, the less 
will be the effect of the acceleration or retardation, and 
the more nearly will the ratio of the lines BC, CD ap- 
proach to that of equality: Hence it follows, that, by 
continually diminishing the increments, the two ratios 
tA and <P may be as nearly equal as we please ; 
and as the ratio ig always between them, it must be 
ates Slats Therefore, the ratio of the velocities of 
time immediately 
stant. 
22. It 
quently ap will be less than 
This is evidently true, not only of f(z)=23, but 
erated with an increas- 
nay be formed of the of a variable quantity 
fom this definition, vet it has been jected by all the 
writers on subject, and some of the 
to in- 
foreten 
' ish: For it has been observ i 
tannins ince Soll tees SLE ccs ies 
F 
fe 
ay 
VOLn IX. PART 1. 
it; that — 
solves itself ultimately into this other fact, that there 
is a certain determinable limit to the ratio of the cor- 
responding increments of a function, and the variable 
quantity from which it is formed. 
We have found that the existence of this ratio, and 
the manner of finding it, rests upon principles purely 
‘analytical. It seems, therefore, to have been entirely 
without necessity, that motion has ever been employed 
in explaining the theory of fluxions ; and although we 
shall retain the terms Fluxion and Fluent, because no 
ee purpose could be answered by exchanging them 
‘or others, we shall not hesitate to reject the cumber- 
some apparatus of ing, as well as the incommo- 
dious notation hitherto employed in this country, and 
adopt the more legitimate theory and convenient nota- 
tion of the foreign writers. 
. Definitions and Notation. 
23, Resuming tlie formula 
+N —F(2)_ 544, 
in which fx denotes any function of a variable quan- 
tity z; h the increment of «; f(«+A) the new value 
of the function when zr to xh; p that part 
of the general expression for the ratio which is inde- 
pendent of h, which is always a new function of 
‘x, deducible from the original function f(x); and H 
the other part of the ion which vanishes when 
h=0; and contemplating the analytical fact, that 
supposing / to decrease, the ratio LE+M—S* ap- 
proaches continually to the ratio of p to 1; we shall 
call this last the ional Ratio.. According to this 
definition, as in the function 2”, we have found that 
(c+h)* —2x* 
—__—— = n2x*~1+ H, (art, 7.), therefore 
h 
Fluxion of (2* ): Fluxion of 2:: a1 :15 
ome eee of Fluxion of | 
‘s): i aieps ls. 
froma which ic flags that : 
Fluxion of («*)=n 2*~) x (Fluxion of 2), 
and in general, fluxion of {rz } =p X fluxion 
of «x. ’ ; 
By this definition, the quantity p enters always as a 
co-efficient into the expression for the fluxion of the 
function; we shall therefore call it the Fluxional Cocfi- 
crent 
From this view of the subject, the fluxion of a func- 
tion is not an absolute, but a relative quantity, which 
depends upon, and is co-existent with another quanti- 
ty of the same kind, namely, the fluxion of the vari- 
able quantity z, from which the function.is formed: 
We pa! yy meng vif maggd peat re of.a er le 
- an i quantity to inde- 
fine ( ele which have to. each‘other the leaiting 
ratio of their simultaneous i 
of this theory, and the claim of each to the invention, 
a i a modes of notation, an in- 
E 
increments. 
The different views which Newton and Leibnitz took ° 
