FUNCTION. 
a=rtAt 224 + ke 
Weak 
esertr toy + = 
+—— = 
1.23 
2.71828 1824. = 4 therefore a4 = ¢,anda=e » 
and when A = Ie-a=it+t1 + 
a. 2 
3 
ta 
xis here fuppofed variable | ; let the difference between it 
and its fucceffive value be expreffed - the fymbol A x 
then i ee eae 
then eS = np Ax poe) y ae 
eh ee easy e* (Ax)? 
+ Tag 
a 
Since e~* =Ir—-wt+ 7 Kee. 
r.2 2.2.3 
* ee. eae _2% 
an rs 2 ees re rs 
& +e* x 
ES at &e, 
°F 2 os rs ree 
eae 
ae es a oe ee eer 
+ & 
ean er tae 4 e-@tAn — (et — e-*) + fe*— eo *) Aw 
A 
++ fe per) © os + &c. 
. And giles at (e —e-*} + (er + e*yAx 
A xf 
+ (e€ He). (: ry : 
— 2 ‘decedent as 
In the form for e* for x fubftitute + «» / — 1, then the 
2? lt _ T. 
fymbol &’—treprefents +2 /—1 
Taz -203 
3 —t 
and é nay reprefents 1 — « /— 7 — —— . ie 
— &e. . | 
, % ly —-:V7=3 
and confequently ao 
+ 
prep a en 
Ee.2 7.2.3.4 
evar ga : oo ri 
‘and = is the fymbol for « — ——— 
5 
+ — &e. 
E.2.3.45 
again et ME y Ae Yo ot YI Aa OD 
aVIn \2 
eae ( 7) &e.. 
1.2 
ee VE ea ME pe VAI emt Ae Wo 
en? Vo (Any 
q. 
eethav—T 4 eta Va 4a + er eV aa 
eee, PARR sacle RTA + atROL 
a 
Pe 
-e 
2 
key 
(A , + ee 
4 &e. = 2.718281824, 
~ is the fymbol fox 
t zWV=1 ent Vt vit ¢ mew —% 
aoe 
I.e2 
The form fer a* being: — fuppofe it were re- 
= to de ermine - m aie 3 n this equat = 
% ee ane an tadependent 
biaryl cuantny,) onfequentty from the laft article, if 
¥(e#—1) + 5 (*— 
1)} — &c. be put = V3. 
then $V 2 + SY)! (Tal + &e. 
=rtaxey AXely + Se), &c. Hence 
comparing the octane: affected with the fame powers of x 
0 VeAKe xa La@o0-ben 9 +% (*—1)}— 
kc. 
— (—1y—4.(*e- 1)? + F(x — 1) &es. 
eis eae 
Hence, if equations, as x = a*, 1 +x = a*, &c. be thus 
denoted, 
x=ah*, 1 +x =ahtt*), we have 
x 
O(i+x)= 
In like manner 9 ( I + 
Ax. Awsm\3 
ss (844 
~ A 
And finee x -- A x = x + € + 
a?* x afa + Ax, 
Therefore $ (x + Ax) = Ox +9 (3 +=*), 
& 
=*), a? @tAs) 
S21 GG - 
A 
x 
or? (x + Any Qe + 
&e.. 
Qn °° Oe 
If « = a?*, eima a, ct=a 3 > Re. 
ee cy 
and generally = m =a@'m ; then. 
Qa q(t —1) — 4 (et — rf + 5 (eB — 2)? 
&e,) 
or = 3 A (eH —B Gah a + Ga} a Be) 
or generally =< ((s™ = 1} ak (a =A + (xm 
—1y- &c.) | 
Tn the equation « = a@*, or *, KX or Gx is ealled the 
logarithm of x, and a the bafe ; . the value of a be fuch: 
that Aor (a—1) —f a (a—~ 1)" + %3@-— _ &c. = Iy. 
ke ative 1.2.3 
and 
— 
reprefenting this value of a by é (= I-i+— 
