FUNCTION. 
gxds 
Fas ae 2 ay 
dx 
axe—x)— fae aoa 
Veen sles) em 
= (a? — 2 x") es ie ea 
dxd(D) = —Satx-#.dx 
mines 980 (etd) 80 off 
d (x ~ a) mda. 
again, in F = 
ad 
3 
N a 
andd (D) = -% oP a OU 88 
This rule is fufficient for the generality of cafes ; and the 
eae continued will give the value of F; this continua- 
tion of the procefs is neceflary when the ae dif- 
ferentiation, ‘and the fubftitution of « = the numera- 
tor and ign: each = oO: an ce of this cafe 
. BV _ 
is ok. (4 — 
If after. ae Can cen N and D each = 0, 
then aia the differentiation as long as N and 
fhall = 
° eae to find the value of £*, when fx, and F x,=0 
on making x = a; for x fubftitute a + (x — a); 
then fx = f flat Ga fatDfa.(x—a)+ 
D' fa. (x —a) + 
mitra F(t (0) =Fe4DFate— at 
D Fa (« —a)?+ 
i ee 0; 
_DfatD* fa. (x —a) + &e. 
DFa+ D'Fa (x —a) + &e. 
== (putting x = a), 
wife = oy and then © = _bfe le D3 fa. (x — a) + &c. 
noes except D fa, D F a like- 
oe Fa («— a) + &c. 
== (putting x = ‘) _ and if D’ fa, DF a like- 
fx on 
wife = o, then Fa = DF and generally, if the terms 
x 
of the expanded feries up to the x’ terms, to wit, D* fa, 
_ fe _ D' fa 
D* F a, = ©, then 7, = a= DrFe 
This rule may be demonftrated thus : a8 a 
d 22 ed fx 
Lety = ye Feo fx. ~Fxt 7 a 
but when x = a, 7 hypothefis ny v= Oy 
d dF x = df x ca } fx 
"IG hdaw 7D. Fee 
which is the firft Lg of the:rule. 
If when F x » DF x , D Fx = allo; then by 
a the erent ‘of equation (1). 
20% Fe+yD. FeoD. f[*3 
a 
bit sine, Oa, ooy= — which 
~ D? Fx 
is the proof of the fecond part of the rule 1, or of the proce 
inued., 
continue 
jt * 
a a 
The values then of fractions, fuch as 
x. 
numerators and denominators vanifh, on affigning particular 
values to x, may be computed by the preceding procefs : 
which procefs is arbitrary, and not neceflarily to be fol- 
lowed, from any thing contained in the fignificancy of the 
f=) . Propofe the queftion feparately, what 
x 
— a” 
whofe 
expreffion ( 
17 
does 
to any mind undebauched with mathematical fophiltry, 
a" —a fo) 
is. OF 2 A. different refult can be obtained by 
= _ become x == a? and the obvious anfwer 
a different procefs: but why ought that eae to be fol- 
No fatisfa€tory anfwer can given to this 
queftion, when it is abftra¢tedly en to find the value 
= 
Rs 
the circumftances under which, in the application of 
» and we can only remove our doubts by viewing 
. . x 
analyfis, it is neceflary to compute i 
Now the fa& is, that in inveftigating the properties of 
extenfio on, © of motion, the nature of the cafe dire€ts us to 
follow, in computing the value of a vanifhing fraction, a 
wl 
procefs exaCily fimilar to that by which on has been com- 
puted ; thus, if y be the ordinate of a curve, and = . 
and A be another ordinate, at an interval (x! — x), and 
or mate 
= 5 then 
between the a where the pias and a fecant of the 
x? es 
Bar “\=4 (x! + x). 
Here x! is greater than x, and the line L approaches more 
nearly to the value of the ag cae the nearer the points 
of interfection are, or the lefs the difference is between x! 
and x3; and the tangent bone. what the fecant becomes, 
when - two points of interfection ae the value of 
(L being a part of the axis intercepted 
curve cut the axis) = 
— x 
ordina — x % 
face is exprefled by the limit of > G —} 
or by what ; (x! + #) becomes, when x! = x, or 
2% 
by = 
Hence, too, it appears, why in finding the value of £* ag 
x? 
L— is put for x and not (w — 4): for fince, in 
ae the { fubtangent or velocity, an expreflion is required 
near to the truth, and. which. approaches to it the more 
e lefs is: the SS between x and a (or x and x), 
Fa 
it is meceflary to.expand * by a feries that converges, 
and it is plain that f ((# —a)+t a) is the fymbol of a 
diverging, ane if . + be — ay of a pet feries, 
Thefe v g fra€tion caufed many difcuf- 
fions cee i vateman cae, ;: , they have ues er faHe 
reafonings. 
