 ‘Fandamen- the num 
i oO rer reve ae 
and denominator of the function 
— errer5 3 os 
+h) =, or, in other words, the terms of the ra- 
or ratio, that is, a ity to which it may 
approach,-so as to afer from 
which we have found to belong to the 
; (pho 
particular case. ——7-— extends to the general func- 
tion C$O"— » being any, constant quantity what- 
é give what we conceive to 
teal mre ng den 
Fonpawentat Prostem. 
6. Let v be an itive quantity whatever, and p 
and q any two whole numbers,. of which g is positive, 
and p positive or negative. It is proposed to 
find two boundaries, between which the function tome’ 
vo? —] 
By Srisip Spee pr 
CaF os Pol op 
ey SFE Oe 
we have a. A 
S 
: 
Fife 
eFeerei 
rr 
£8 
fal 
be-expressed exactly by po",* and so we shall have _ 
FLUXIONS. 391 
Pp 
ae and’ 1—v"= p(i—v)v®, (1) 
Tn considering the nature of the quantity P, it ap- 
pears to have these four properties : , 
1. Its value ds on the value of p, and also on 
that of v, so that it is a function of p and v, which we 
suppose to be independent of each other. , 
2. It is always greater than 0, and less than p. 
3. If p be supposed to increase, then P also increa- 
ses. 
4, The quantity P increases slower than p, so that if 
p is increased by an unit, P will not be increased by so 
much ey unit. Bi ‘ 
_ The two properties are sufficiently evident. _ To 
prove the third and fourth, let us suppose that when p 
increases to p+-1, then P becomes P’, thus we have, 
pr =1tvofpr?... 47, (to p terms, } 
(p+ lv” =1-+040"... +0? 40?, (to(p+1)terms.) 
Let the first.of these two series be denoted by N ; then 
we have evidently . 
pe =N; (p$1)o” =N4+v?=1+40N; 
and hence, dividing the corresponding members of these 
equations the one by the other, we at 
(p+1)e—" i id (2.) 
=f +-pv 
Now we have evidently 
po? _ oP 4 oP 4 oy &c. (to p terms) 
N Fp ov pet er pee 
— Ba1+14+1+4+ Ke. (to p terms), 
Teva... ae 
| Then, as v is less than unity, the. numerator of the 
first of these two expressions is manifestly less than its 
denominator, and the numerator of the second is greater 
than its denominator ; therefore, the first fraction is less 
than unity, and the second greater ; so that 
P 
t= 1, and, PS; 
and hence, from equation (2), : 
(p41) 2 ee Pot 1, and (p-+1)0" =’ S14pv} 
t v being les unity, 1-+-pv>~v +-py, 'that is, 
1+-pv > (1+p)v; therefore, also nes 
(p+1)0"—" >(+p)v. 
expressions it appears, that 
vo”? = 1; and v” —"Sy; 
and hence, by multiplying the quantities on each side 
of the sign of inequality by »”, we have 
o” =v" ; andv® yt, : 
As v is by hypothesis less than unity, we may conclude 
from the first of these expressions, that P’ "is greater 
than P ; and from the second, that P’ is less than P+-1. 
it appears, that the new value of P, which corre- 
to p+1, is greater than its former value, but 
thebts Setiee exceed its former'value by. so-much as 
From 
i*\Here we take for granted the obvious prinéiple, that if a 
‘ : variable quanti from one state of i i a 
coming Dp ob a abi q Agua magnitude to another without be 
og infinite, it must successively have 
5 
Fundamen- 
saleoes 
—— 
