392 
which we proposed to demonstrate. 
Since it from Equation (1), that hemp ie 
any positive integer number, then 
1—v? = p(1—v) 0, 
So also g being any other integer number, 
a corresponding quantity Q, such thet 
1—vi sg (1—v)v* ; 
waere et ae 
P has to 
app poses sh any w 
ae af 
, for an instant, that 
Pad Q Q will also be equal, 
org. Now, let p be 
reas but as it 
number, there must be 
relatively to q 
dividing the 
other, we get 
(3.) 
pi poe 
to increase 
not increase 
we hine POs (pd): As is on Pand Q. 
snd these again on p,q and'v; it follows, that zis a 
function of p, g and v, It is =0 2 ens 
every other case it is a itive than uni- 
ty. By sbattating ¢ (pg) fo P— for Posh wa habe esr 
I—v? P yi PD 
dawnt +, qd 
Let. v=— = #80 that » being Jess than unity, 1 will be 
greater than unity- Instead of v, let > be substituted 
in the formula just now found, and it becomes 
th P nI—) (—9- 
ufl—1 
As zis some 
has been investigated upon the hypothe tat ls 
than 1, io aloo trae when et greater then 
Pin we positive Guksley 
We have hitherto su 
but to include in the nla thecassatp “oe 
tive quantity, let both sides of the oquetion 
plied by ov”, and we have * 
1-0? —p Poe 
I—vi q 
Assume now (p= q) —p=# (—p—9), and'then 
= -— : hence, as all possible values of z 
iPr 
included O and 1, so, all possible values 
ovis damaanee DXB DP ane 
13e( eae 
= th llomearents 
3, and so we have 
FLUXIONS. 
Pentmeeh 1, and these are the two properties of the quantity P, 
= a a Vy" 
sy ten 0nd br 
niet, instead the second mem- 
— the elation wil hae the same for a etre 
w , then, it » 
penton: i orhatever, sand p and q 
numbers, of which q is positive, and p oUber ploy or 
(rh =a, ond, fe EO 
4 Hiren 
a—l / / 
to ee j Naettibie 9 ait oS thera 
q » 
cr = mn 
and —— 
ves ; 
ese os 
ne, ti 
, "ut anne be bee 1 
sf Ot | ad at oP te. genta ‘expan 
ne the ratio, betveen nem, soda fa = ) 
hen (24-8). 
rive Soria alo to's: Blah Olin Siete 
the expression for the ratio approaches as 4 decreases ; 
— 
