$94 
Feedasiee- Henee p and ¢ may be any two whole numbers, as 
wl duit he 
—_—o 
as we please, ve to each other the ratio of m log. x 
to w log. 6, which indeed is incomm , but may 
be expressed by numbers as near to perfect accuracy as 
vwe choose. 
The remaining quantity » will be found from the 
= ; 1 
Log. = log. x; or this, log. o= ag OB > 
Now as by hypothesis r=b", and by assumption z=v?" , 
and 6*=e"?™; therefore o?"=u"?, and hence pn=ugm, 
and 2-=™™. Also, sincev”=2*and o! =o, therefore 
q 
et =. Let these values of »”, »’, o°%, and F be 
ba 
substituted in the formula 
1 
=Ly—2, 
vi —1 q 
which was investigated in Art. 6. and it becomes 
i z : 
eal aes 
bu Ps 
and hence, 
Od scowls (C) 
a ™(b"1) 
where m and mare, as we have already observed, quan- 
tities of any magnitude whatever ; and z is some quan- 
tity, mp -of an intermediate magnitude between 0 
This formula exhibits elegantly the boundaries tothe 
value of w ; for as z can never be so small as 0, nor so 
great as 1, if we put ==0 and z=1, we shall obtain 
two expressions, one of which is greater, and the other 
less than v. These boundaries are 
z AE ALS 
n(x —1) _— 6” n(xr—1) 
1 x é 
m(b"—1) x m(b"—1) 
They are remarkable on account of their involving two 
arbi quantities m and n, which have no apparent 
connection with the function they serve to express. It 
also appears, that the bounding values of the function 
oe 
are to one another as 4” to x” ; now as we are at liberty 
to give as great values to m and n as we please, this ra- 
pater Breve: degree of nearness to a ratio of equa- 
er ence it appears, that the quantity « is a limit, to 
w 
. 
is equal to itself. The equation 4 — 
; v1 | ol 
ber to a common denominator. An identical 
FLUXIONS. 
supposell os incest and to which come near- Fandamen- 
er by an par a A ane us we have _. tal 
© complete solatiant at a 
10. If we suppose 6 to be the basis or radical num- 
ber of a system of logarithms; then, in the equation 
«= 6", u is the logarithm of the number x. (AtcEsra, 
330.) We have, therefore, from | C) this 
~_ expression for the logarithm of a : 24 
= I rE ede 
b” n(x" —1)° Lt 4 
log. 2=7;-"™— @). 
a m(b" —1) Bis, 
being some quantity contained between 0 and 1, and 
‘i and m any readied Ghaaorer: 
In effect, therefore, we may consider, that 
L 
log. 3 ol Re 
(E) 
m(b™—I 
vided we do not limit the magnitude of them and n, 
but them as greater than any assignable num. 
bers. Under either of these forms the expression for 
the ithm of a number is valuable, because it iden- 
tifies logarithmic and exponential expressions with coms 
mon gion quantities. 
11, 
erhaps, it may be doubted whether such an exe) 
L L 
pression as n(x”"—1), or m (6"—1) admit of any defi- 
nite value, bapa BPR of dee lndefiagende of the at a 
mandn. To remove this difficulty, we shall resolve 
this third 
ProBLeM. 
Let v be any positive quantity, and n any very great 
number, or ra‘ a mantioer preheat pf 
number: it is proposed to transform the expression 
n{~"——1) into another that shall be free from the inde- 
finite quantity n: and also to calculate the value of the 
expression, in some particular case ; for example, when» 
v=10. t n 
Solution. —Let V and V’ be two functions of v, so re« 
lated to each other, that 2V"=V+41: Then 2(V’*—1) 
=V—1, and pes A ae 
eg VideL it Veeed 
its two boundaries and the intermediateexpression as _will ap’ by bringing the terms in the second 
= I member of the equation into a single ion. Hence 
a(x? —1) A we have this identical equation. * pT IN: 
oas\* 7 continually approach, as m and n are V+i 1 Ve 
a mI). vam Hyqitt 
® An identical equation is so called, because it may be changed into another which shall epress merely that acertain quantity 
Pons is of this kind, as will appear by reducing the terms in its second mem- 
equation differs from a common algebraic equation, such as v’—3v+2=0, in this re- 
spect, that the latter holds true only when v : 5 ; : 
has certain particular values, which, 
other bolds true when v hay any value whatever. Many 
etrical theorems, when expressed 
“equations. The fourth pryposition of the second book’ of Kuclid's Elements produces this, (x-+ 
in the present case, are v=1 and v=2; but the 
‘braic symbols, are identical 
=a? 2ry--y*; and the ninth 
‘2 2 
sand tenth produce this other, rt yt = aL ace Lin theme end y may have any magnitude whatever. “ye 
<3 
te A 
