Se 
and = 2. Upto &e.) 3 then in the equat 
e?*, X or Ox is commonly “called the hyperbolic gt 
ecaue it can be refented by the area equila- 
m 
t has appeared that 
‘ : — I 
(2 + Ax)" = x™  me™~ bax + ” nie Op 
(Ax)" + &e. 
=a" 4+ Dx™.Ax + al x™ (Ax)? + &e. 
. v, A? A ? 
‘Likewife that a*+4* = a*.A.Ax- eee, &e 
A 
ettAr = et peax ge? aa + &e. 
and that in the equations x ++ Ax= ee, x+Axr= 
t@tAz). 
3 
Ax Ax : 
Ax.3 
(=) —.&c.3) 
x 
Ax Ax? , , Ax} 
Het dn) alae — 4S) +8) 
&e. 
In the above expreffions L . x denotes the common, and 
1, x the Napierean or Eye boue Deane e of x 
e expanfions of the forms x”, a‘, e*, 1, xy when 
x is ne to (x + A-x), it appears ee that there are 
certain common properties whic redicated, 
it aa an an of ana shape concern ie tees chete 
expre mon fymbol and name, 
and to ‘ompreten yee auton under a gener: ao 
mula. 
La, or / xf Ax 
Now the Paar D i ee only denoted that ope- 
he co-efficient of the fecond term in the 
Aa)" is clans Let it 
-made on x”, a‘, 1.x, 
&c. ee pares the co-efficient of ie fecond oan in the 
sal area of ¢ 4+ Ax)” (a°t4*), Li (m + Ax) (x +4e@ 
s formed then under this fignification ; its 
foeacs one will be comprehended, but it will no longer 
denote a eee Age as when reftrited to expref- 
fions fuch a iia when applied to x, gene- 
rally veprefenta the (pee term of the feries that arifes 
ag ren g ox, when for x we fubftitute its new value 
and that fecond term can always be known 
ae “il the e expreffions which ¢ (x + Ax) is made gene. 
rally to reprefent, have been previoufly expande 
Thus, De™ omx"™"', 
2 - Da — De? = e 
1 
DL.«= a Dl.x= x" 
» 
Hence D D: x® or D* x™ denotes m. (m — 1) x™~", 
D Da* or D’ a’ pig poe a) & a - 
D D Da* or Da” denote 
Again, DD /.x or D*/. x denotes D x or — cm 
; - pe 
2 . 
OY 
x 
DDD or D'./. x denotes D (D*l.x) or D(5) 
. In like manner D‘/.x denotes D (D*/. x) or — 2-3, 
. 4 
Hence, fince 
(4 + Aw)" =a™ + me™—" Ax + &e. 
A? a* c ey 
atéz = at4+ Aad. Ant + &e, 
A A 
Li@+Asy=a lL. e+ 5 , Oe ee 
motte 4 Beye tent BE 4 CS) + &e. 
under. the follow 
Q(#+Axz) = Oe + 
It appears a ail thefe forms may be comprehended 
ing: 
Dew. Aux + - Gx (Ax)? + &e. 
ray 
or=%4+ Dex. Au + Dee oe + &e. 
which agrees with the theorem known ‘by the name of 
Taylor’s theorem, and given by that learned mathematician 
in his & Methodius Incrementorum,’’ wherein he fayst that z 
flowing uniformly and becoming x 4. v, « becom 
v - aa? 
re oe = + : a + #.——- 23 + &e. 
i L i2s 
The cfc of eos — third, &c. terms, are re- 
prefented by D¢x, D° > x, D' @ x, &c.; and when > # is 
of the form /. x, be == e! x) — the firft term is /..%, and 
the co-efficient of the fecond i is — ; but — = the co-efficient 
‘ I P 
of the third term, is not formed from — as — is from - +. 
When D 0x, D? $x, &c. are faid to be derived from 
each ota by the fame ‘law, it fhould always be previoufly. 
defined, that by the fame law is meant that which orders 
the fecond term of the feries to be taken when the fun¢tion 
of «, whatever it is, has been expanded after fubftituting 
w for x. 
e gt (Ft 4) or et FA) will ikewile be included 
ar ie the form @ (w# + Aw) = Gx + + D 
he x & wv)? + &c. it is tea bases if ¢ (w) be made to repre- 
* + e—*, that —Gt4©) will likewife be 
ane under the nee “foun ‘0 (@w + Ax), and its ex- 
panfion likewife under the fame form as that of 9 (w + Aw.) 
Again, if the fymbol / — 1 be vais ¢ (EQ@TAN SAI 
may be included under the form @ (# =oe+D¢ 
5s x), and sconkequeatly if Ox be 
made to aa ev + e-/—'; then echt sar + 
T@tAyV —t will i cys under the rae : (2 + Sx) 
=90+Dox D: . 
Tie fay 9 4 Hy 
tad refent the cofine and fine of an arc x, 
he Sai 
whofe radius i is unity, and confequently with reference to 
their 
