FUNCTION. 
It was not perceived that, to affign the value 
- («= 
reafonin Bs. 
a), there was an abfolute mans of 
finit tion, convention, or extenfion. The n of 
effentia — ing 
of <= 
fome de 
an chee fignification, and of an 
8 
we 
to fvch an exprefficn, as 
valued themfelves on the clearnefs - their apprehenfion, 
and the jufinefs of their inference 
The method of limite, or of opie and ultimate ratios» 
Landen’s methed, the method of finding the value of = 
(fu, Fx ) are related methods: they all demand ae 
fame thay affumption, which has never been exprefsly 
made ; they are all equally ety to an objection, which has 
never been fatisfa actorily removed. 
Such is the theory of thefe nada ae given by Mr. 
Woodhoufe, and which is fo clear and f. tisfatory as to 
e ana- 
lytical funGtion of x, @ 
expreffed by this ae @ x 
“@ «x (Ax) + &e.3 D as the note or: an operation 
by which the eaeetie ont of the fecond term ex- 
panded expreffion is cbtained; thus if ¢ x rosy 
reprefents x”, a, e,/. 43 D % « reprefents m x’ 
Aa, e, : ; now in the expanded expreffon of ° + “ 
x 
for A x Ea ch fymbol oS then ?(@+ dv) = 
Pw. dx)? + &c. and let d ce that 
eperation by wie the fecond term of ¢ (x + dw) is found, 
which cafe it muft me pee firft term = are entire 
difference of @ (a + 
dex 
Dx .dx, confequently D ¢4 = a 
D¢ x), d x expanded 
d.(d¢x) 
(da)? 
a (employing d 2°, dx’... d 2” to reprefent (d a) 
3 again d (d $ x) 
== firft term of (D 9 (x + dx) — 
= 
= D’¢a.(dx)’, confequently D? Ox = 
d 
dw 
((d )* cressenns a ( By ba d vi ee ord’ ¢x#= D 
(Vow) d / =Di¢z.dw; 
iuety Di@a= 
a" Qe 
are 3 and generally D* ¢ x 
xv 
The expanded form for ¢ ( 2+ Aw) may ther be 
= 
= 
e 
a 
thus reprefented ; 
—— 
da 7 (Ax) + ec. 
Ret area 
and fimilarly . 
“Oe 
doe 
oof Gok de 
a 
a ee 
C(wtdae)= Oa + 
The difference between ? (w + d 
entire di as and its note or ebe) is Ag 
(v+dx) —-Or= ses "eos 
of this entire difference, viz. the fecona term, is called the 
a) and % x is called the 
thus A @ xor 9 
A part 
rde ty + &c. 
differsntial, and its note or fymbol is d; thus d (@ 7) = ee ' 
Vou. XV. ‘ 
x + ax) expen may be: 
baad 
dx,or DOx%.d4%; the fecond, third, &c. nt th terms of 
the "feries for A Q 2 (abit ing the numbrical part of, the’ 
co- ria are called the 2d, 3d, &c. ath differentials ; ;: 
tou 
dow d'o dOx 
A@e=aw i eae —— 
Ox = dz wae re aa oo od a 
€ ae » ¢€ aE 
+ &c. ac .dey da ee. 8 = . dx” are 
the fecond, a nth, &c. ee als. 
Since ‘A is the note of the entire difference, and d of the 
firft term of that difference, when thefe notes are applied to 
a fimple quantity, as x, they are eqaalg fiznificant; thus A » 
d isa aoa 
oF es da it rere | 
do 
calculation; thus — ” denotes, hats in @w, fore, «+ dx 
=dwxr. &c, are to be confidered as types of 
is to be pu*, and the co-efficient of the fecond term of 
@ (w +d.) expanded to be taken ; : 
called the differegtial co-efficients. 
A few inftances will fhew the fignification of iy fymbols 
4, D,d; 
Fe mn a8, (Ax) 4 &e, 
| hen Ag t= (#@ + hase av", or = matt, dx ae 
men ~1) 
~2 
Axv™ =m ae"? , 
ad —2 
da’ + &e. 
when = te + dat a 7 
Dx » fince D x” denotes the fecoud term of 
Se a 2) expanded ne pa 
3 v™ is aacladed within brackets to pre- 
on ambiguity, hae d oe has already been ufed to denote 
(d x 
Aa 
Again, if «v + A» be put for +, Aa = Aw 
«(A x)? + 5 
Ava 
if eo + ee forv, Aa’ = ae. da + <a (d x}? 
+ & 
Da 
= My an =e 
ret low Ping. dependant on dee equat ; 
Al.« = (the difference between two eee ene oF 
ra aa Se aoe 
—.(F “y 45 (AZ) —— &¢. 
‘Die =e 
= (fecond term ave : i eae =A a, d (a). 
d(/. ak, 
Gener rally, if y be any fun@tion See igi, diay +Ay 
be what y becomes, when » + Aw is put for wv; then 
ay=e let as) — pea Doe dst Pee. (Ax)* 
+ &c. or as ++ 2 (aay + &c. 3 in which . 
dy 
formula is a2, &e, ate to he confidered as types of cal. 
ed denoting that inya funtion of ay” + dw is 
“3L 
culation, —— ie 
te 
