ioe 
OOD, DFO SEE rn en 
2s 
, 
~ 
AN ae En Rt. 
FLUXIONS.. - 
Am 2 4B mix” 4A mgmt debe.) ht. iene --Yese 
for then the fraction =) approaches continually to 
unity, iid a: the bowidnnies 4 the value of the ex- 
-g(w—1) remain unchanged, the function 
fete ee will continually approach to unity, and 4 
the ‘expression for the ratio to n 2*-*, which is 
a limit to its value, because the expression 
ft =~) nay differ less from unity than by any 
pe ey ck x 
have seen for the ratio is al- 
cart toeponarersead 6 lea These two ex 
pressions may be included in this one, » {(x-+-h')’—?. 
where h’ is put for a certain quantity greater than 0, 
and less than hk. We may therefore express the ratio 
otherwise, thus: z 
en =n (sph) (B’) 
end in arder to fave the two bo boundaries «ph tige 
we have ve to h’ its two bounding values, viz 
Oand h. This & another another formula, which shews also, 
in a very distinct manner, the properties of the expres- 
sion for the ratio. 
From this last formula, we have 
ch) n (24h) "th; 
and as, 
fmm lm mE 1 h’, 
on ta (0 hye, | 
inne and 4; we have, by pila 
Teeeiae ats 
ac Paarfns—ht-o (nt) (24h? hh 
ae = — natn (nl) (ch). 
This expression for the ratio is composed of two 
parts, one of which, nx*—", is entirely independent of 
the increment h, y and the other m (n—1) («+ e”)*— W 
is a function of » and hk, which can tly never be 
infinite while / has a finite value, but which vanishes 
We may denote thi smn eg ce 
simply by and then the Rito may Vis cowie the 
CA ane 
ate ei teary ld Sega cos prop dos Of te oe 
= Tae 
tio may likewise be detuced: - 
(c-+hy'=a" +n27"— h4Hh. 
which we have demonstrated to be- 
~ est ead cg us to prove that every 
function whatever, - a variable quantity 2, 
w=A (c4-hy? 4B (x mt 4. & 
Then because ae bat ips ns G(r te 
oe), as Fac Hh, and ha 
z = My so on, Wi 
art a Bd PO ne ete 
taliyaiecHt.&e. + &e. 
CH'"h 
pe Cones + AH 4. BH” clin &e. which vas 
nishes h=0, be letter H, 
then putting « for Ax® 4 hak oe we have 
VOL. IX. PART I, 
393 
a'=u+ 
stp any hence 
= “=A ma" 4B m! 24. &e, + H. 
Here it appears that the expression for the ratio 
— has precisely the same property as has been found 
to belong tothe ratio @F")"—=", it is composed of 
two parts, one a function " x only, and the other a, 
function of x and h, which vanishes when A=0. 
The fractional function 
Az" 4Ba™+4. &c. 
aa" + ba” + &e. 
has the same ; for by the substitution of x / 
instead of z, numerator becomes 
Ast4 Bov4-Se-4-(mAz™ +m’ Ba" —'4&e.)h+ H’i,. 
and the denominator becom 
ax" +b x” 4. &e. 4 (nae nb. 29'1.4. 8c.) hb". 
Put Ac™+-Br" 4 &c.=N, 
mAz™—1 +m’ Bo”! &c.=N’ 
ax*+b2" 4. &e. =D 
nas + nbz" +4. 8c, =D’ 
and we have 
eae wae Lote 
~D+Di+H% 
This expression, by actual division, is easily transform~ 
ed to 
ok 
Sp. ied pan Parmete capone, Wiis wh tncienntorbnely 
ba the quotient which is by one 
other of the three quantities h, H’ and ” ann which, 
consequently, vastishes when A=0; therefore observing 
i= 
As the tities D; N’, N and D’ are all in t 
of h, it 1s manifest that,” bs as capo, in veto es the 
ese espn ns gg Peres a Dong oF in 
9. Nahas ona bert at wtem ic ohiedgacet = -anezhal 
logarithmic exponential quantities, according to 
the plan we have laid down, c requires that we resolve 
this other 
PRopLem. 
which is 
Also let « be such a fanction of «, that c=. cand 
posed to find two boundaries to the values of w ; that is, 
two expressions, one greater and the other Jess than w. 
Investigation.—Let m and n denote any two given 
tee yaad ied iy, Ge beth grocer coast oak 
tities « and 4, provided are both greater uni- 
ty, it wil always be possible to find two whole nut 
bers p and q, and a positive quantity v, such that 
= a, tobe 
For, in order to determine them, we have by the theory 
of ; 
1 1 
p log. v =— log. x, and q log. v = 5, log. b. 
mlog. ¢ 
qo alg t 
Spd 
Prineiplén 
—_——— 
