Pee eZ q 
Ce gat 
ee 
FLUXIONS. 
second degree, y will have two values corresponding 
to any gi 2 of x, and as y enters into the ex- 
pressions for the fluxional coefficients $4, 4, fe 
these will also have each two values. A like remark 
will a ene ys ye coefficients ei eh an 
primitive equation of any higher order ; number 
of values of y determining in every case that of the 
values of the fluxional coefficients mto which it en- 
ters. 
49.. We have seen (art. 27.) that the constant 
quantities. which enter into a. function of a variable 
Emagen B athnatare 1.0" i from their fluxions. 
he same remark applies also to fluxional equations. 
‘For example, if 7?= a 2+ 6, the fluxion 2ydy=adz 
belongs to every particular equation which can be form- 
ed from the equation 7?= a z+4, by giving all possible 
va he flucional be also expressed inde- 
uxional equation may be p i 
pendently of a, by eliminating this quantity by means 
of the two equations 
i dy _ 
youth, T= 
yoy 
. , igi 
we then find 
dz Qiy Le, 
This equation expresses a relation that subsists among 
the quantities 2, y and %, independently of any part- 
cular value of a. 
. If the constant quantity which is eliminated is\not 
of the first degree in the ation, the result 
obtained will contain powers hi the first of the 
fluxional coeficient 4%. For example, let the equation 
be 
Hence, by taking the ftuxions, we find 
ydy—ady+2dz=0, 
z 
therefore a= YY SES, 
This value of a being substituted in the proposed equa- 
Pe an: 
(z*—2y) WY — gay —r=0. 
This equation expresses the relation that ought to sub- 
sist between the variable quantity x, its function y and 
its fuxional coefficient A independently. of siny parti- 
cular value of a. 
By resolving the equation y°—2 ay +.2*=a*, in respect 
ofa,wehave — 
a= — yt (2y*+2°). 
As a is now pan pages le quantities, it 
ya en in taking the fluxion: accordingly we 
a 2ydypede 
4 — y= 78 ay = 
‘ When this expression te freed fori the Tadical sign, 
it will appear to be the same as we have found by eli- 
50. number of conten quantities whatever, 
contained in an equation, may be made to disappear, 
—— fluxions as often ge thata ape questing 
Let y= m(a*—2"): by taking the fi 
” f 1 E 
2ay+2*=a%, 
we find 
i 
411 
dy , Pp D 
deme Taking now the fluxions a second y 
time, we get y SY 4p Pm; this value of —m 
being substituted in the former equation, it becomes 
oy td =o 
Vis * Ia —*9 Gam 
a result which is independent of the two constant quan- 
tities m and a. 
Investigation of Taylor’s Theorem, and its Application 
to the DracUipedent of Functions. i 
51. The principle established in art. 28, and. il- 
lustrated by various examples in art. 36, 37, and 38, 
leads immediately to an im t application of the . 
fluxional calculus, namely, the developement of func- 
tions into series. 
Let us consider, in the first place, the particular func- 
tion 2", x being su variable, and nm any constant 
quantity. We have found (art. 7.) that when 2 be« 
comes x-+-h, so that 2" becomes (2+4-4)", then 
(xh)" = rp leer +H h, 
where H is a function of x and h, which vanishes when 
h=0. As x and h are quantities which we suppose 
to be entirely i of each other, this is an 
identical equation of the same nature as the equation 
(z#+h)}3 = 23 ft Ow" + 32h + h*)h, and will hold 
true, whatever values we give to « and hk, We may 
therefore, instead of h put 4—x, where / denotes also 
a quantity ind t of x By this- substitution, 
(oth), the first member of the ion becomes 
; but as the form. of the function H is unknown, we 
cannot actually make the substitution in all the terms 
of the second member ; we may however suppose, that 
if it were made, the quantity n2"—-14H would be- 
come X, a function of z and £, and then the equation 
will be 
kt = x" 4 X (k—2) 
an identical equation involving two indeterminate quan- 
tities x and k, which, being quite independent of each 
other, we regard k one of the two, as constant, 
and still the equation will be true whatever values we 
give to x the other quantity, which may be now consider- 
ed as alone variable. This equation may therefore be 
treated exactly as the identical equations we have con- 
sidered in art. 36. that is, we may take the fluxions of 
all the terms, and after dividi dx, we shall have 
‘a new identical equation; and this equation may be 
treated like the former, and so on as often as we please. 
Accordingly, taking the fluxions, idering & as con- 
stant, and observing that d (a”) =n a"—1d x (art. 26.) 
and that d {x (i—n)} =(k=2)dX— X dx (art. 29. 
and 31.)-we have 
=na—'dx—Xdx+(k—x) dX, 
and hence, dividing by d x; and transposing X, we get 
X =n 2-1 4+ (k—z) 
da’ 
‘a new identical equation, involving x and k, We next 
take the fluxions of the terms of this equation exactly as 
before, considermg 4 and dz as constant quantities, 
and get a 
dX=n(n—1) 2% 2*damdX+ (k—e) z ye 
And hence we find, after dividing by d x, 
