COPY NO.



NEWTOKKlNVERSmr

lfgnnJ1EC -. ca»cw



MAR 30 195:



IMM-NYU 215
MARCH 1955




NEW YORK UNIVERSITY
INSTITUTE OF
MATHEMATICAL SCIENCES



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* W,_ . Ufi, ^ 17CU science.



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Growth of Drops by Condensation



° Ce > N ew



York 3



IGNACE I. KOLODNER



PREPARED UNDER

CONTRACT NO. DA-18-064-CML-2518

CHEMICAL CORPS, U. S. ARMY



Hy;



Hi

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IMM-NYU 215
March 1955



NEW YORK UNIVERSITY
Institute of Mathematical Sciences



GROWTH OP DRO^S BY CONDENSATION



Ignace I. Kolodner



This report represents results obtained
at the Institute of Mathematical Sciences,
New York University, under the auspices
of the United States Army Chemical Corps,
Contract No. DA-l8-o6ij.-CML-25l8.



., I:



Growth of Drops by Condensation

by

Ignace I, Kolodner

When the vapor pressure (or concentration) in air sur-
rounding a liquid nucleus exceeds the saturation pressure for
a given temperature, vapor will condense on the nucleus and
the drop will grow. The object of this report is to determine
this growth. The problem, although of great interest in it-
self, is preliminary to the much more difficult study of
behavior of a collection of drops in supersaturated medium -
a situation which may serve as a model of the formation of
clouds.

In Section I the problem is formulated. Section II con-
tains the discussion of results obtained here in comparison
with certain results already available. The remainder of the
paper is devoted to derivations and proofs. Many proofs are
merely sketched*

I. Formulation

In order to determine the radius R(t) of the drop it is
necessary to consider simultaneously the concentration of vapor
c(r,t) around the drop. If we assume that the temperature
remains constant and the drop remains spherical, then c(r,t)
and R(t) can be determined by solving the following problem:



I



r;



â– 



•'






•












'



.



'••; .


















..



'



.
























■•






'






.



1 ' '.









(1)




c * - c = c. for r > R(t), t >
rr r r t '

c(R(t),t) = 1 for t >

ac (R(t),t) = R(t) for t >

c (r,0) = for r > 1

c(co, t) = for t >
R(0) = 1.

In equations (1) all quantities are in the diraensionless
form given in [1] (see equations (7) - (12) in [1]). The con-
nection between these and the actual variables r, t and con-
centration c is given by:

r = ar

- a 2

(2) t = %- t

c(r,t) = (g - c Q )c(r,t) + c Q

while

g - c

(3) a=- °- .

p - g

The meaning of symbols used is:

a - initial radius of the liquid nucleus

D - coefficient of diffusion

p - density of liquid

g = g(T) - saturation concentration at a given temperature

c - initial concentration of the liquid's vapor.






■•



1



- 3 -



In the problem under consideration, g < c , hence a < 0,

c o - g

On the other hand, since p > c , a = - — > -1, while

' r o' p - g

a = -1 only when p = c , that is, under critical conditions,
in which case liquid cannot be distinguished from vapor. (As
it turns out, the mathematical problem as posed has no solution
for a < -1 in accordance with the fact that no physical
situation leads to such values for a, ) For water droplet in
air, at 73°F> g = 2 X 10"- 5 g/crir ; hence, assuming that the
initial concentration is twice the saturation concentration,
a = -2 X 10 o In general, a is expected to be of this order
of magnitude o



II, Review of results

While the corresponding one-dimensional problem can be
solved explicitly, only approximate and hardly justifiable
attacks were known to apply in the three-dimensional case. The
simplest among these is the so-called quasistatic approach which
uses the assumption that c(r,t) is slowly varying in time.
The term c, is then dropped from the differential equation,
thus forcing a solution of the form

(k) c(r,t) = f(t)r _1 + g(t).

From the condition at infinity we then get that g(t) a 0,
while the first boundary condition determines

(5) f (t) = R(t).















• •






























[



•-












•












• . ;



- k -

On substituting in the second boundary condition, we now get

(6) R = -aR" 1
or, on integrating,

(7) R 2 = 1 - 2ato

In this method, the initial condition on c is violated.

A second type of attack which is in fashion with physical
chemists, consists in assuming that the boundary moves slox-xly,
so that it could be at first considered that R = 0. (This is
equivalent to the assumption that a is small and the method
ought to lead to the beginning of an expansion of the solution
in powers of a)« Then R = 1, and the system (1) less the
second boundary condition is solved with R = 1, and the result
is used to get an improved equation for R(t) by using the
second boundary condition. Thus we get



(8) c(r,t) = -?—

V\pt



co



r-1



-CT 2



Z\fx
from which



fr-l\ 2



(9) c r (r,t) - -*-/$ J e^ do + -i- e ^

r VTr | r - V-l 2 -/fc

2Vt

If we set in (9) r=R = l, and then use i _ Bo2> we will

get a result too much at variance with the quasistatic solution



i â– 



,



r • •.••■;. .



i















t



.






.



- 5 -

(7), namely,

(10) R = 1 - at - — Vt .

y"F?

If on the other hand, one does not specify R in (9), one gets
on using l _ Bc2 a complicated differential equation to solve.
Of course one can force an asymptotic agreement with the quasi-
static solution by replacing in (9) r by R(t) outside the
parenthesis, and by 1 inside. For then one gets, on using
the second boundary condition,

(11) R 2 = 1 - 2 at - — Vt .

It is usually argued in favor of formula (7) or (11) on
the basis that experimental data predict a linear growth of the
area (i|TrR ) of the droplet. But this by no means insures that
the mathematical theory is sufficiently correct for purpose of
prediction, .since the available experimental data need not
cover a sufficiently large parameter range, while the mathe-
matical validity of these formulae is doubtful. (They were,
indeed, almost forced into agreement with an experimental
formula).

Another deficiency of the methods just discussed is that
they are difficult to iterate to get even formal corrections.
A better, although still formal attack has $een devised in [1],
where formula (11) was arrived at, accompanied by a statement



- 6 -



that the error is of order of V^«

In this paper the problem is solved rigorously. Although
complicated and powerful techniques had to be used, we get a
simple practical result . We find that for {3 = -a sufficiently
small, there are three numbers, A, B, and k > 1 such that for
all t,



(12)



A + JL < RR < it(A + — )

Vt - yt

,2



1 + 2At + 1+BVt < R < 1 + k('2At + \\3*/t) .



Thus, both RR and R are determined for all t within the
relative error of (k - 1). For (3 < .05, a crude estimate
for the constants is given by



(13)



A
B



(3 = -a

A e -P



k = 1 + lu32Vp,



For the case specified in Section I, formula (13) gives

k = 1.0122, showing that formula (12) determines the radius

with a relative error of .0122.

As already stated, formula (13 ) gives only a crude esti-
mate. In Section v we discuss a simple numerical procedure
of obtaining sharper estimates. Thus in the case considered
(B = 2 X lO"-') one can show that while A and B can still
be taken as determined by (13), k is reduced to 1,00563. For









-



:



'


















•



'



.









- 7 -
larger p, however, say p = .05, while (13) yields

A = .05

B = .0267
k = 1.96

the method discussed in Section V leads to a sharper result,

a = .0525^ = 1.051 p

B = .0281 = 1.052 -£- e -(3

Vrr
k = 1.338.

Incidentally, these considerations show that formula (11)
is, after all, quite satisfactory, and that the assumed mathe-
matical model for condensation predicts a linear growth of area
of the drop. While a correct asymptotic behavior is predicted
the deviation of the approximate result from the actual solution
may be considerable for p > .05.

III. Condensation problem vs. evaporation problem
The evaporation problem discussed' by the author in [2]
is formally identical with the condensation problem, see [2],
p. 1, the only difference being that in the former a > 0, while
now -1 < a < 0. This seemingly small difference is sufficiently
important, however, to invalidate the main results of [2].

Let us briefly review the method used in solving the evap-
oration problem. We first constructed an auxiliary function,












.


















- 8 -
eq. [2]-(l8),
(111) uP(r.t)-ir* £l^ z . (1 + a-^pCcr^cT) ♦ 1 (g)

2 J o t " ff yt - o '

defined for -oo < r < oo , t > 0, r^p(t), where

(15) z = z(r,cr) i r - P {
2VF- a

(16) r.(z) = i exp(-z 2 ),

1 vTt

while p(t) was an arbitrary function. If p(t) were the
actual evaporation curve, equation (l!+) would yield rc(r,t)
for r > p(t), and would be identically zero for r < p(t).
We next observed that if R(t) is the solution of

P

(17) Vp~,t) = 0, p(0) = l,

and if in addition R < 0, then

(18) u R (r,t) s o for r < R(t), (theorem [2]-3),

and as a consequence, u (r,t) for r > R(t) satisfies all
the conditions required of rc(r,t), (main theorem), concluding
that — u (r,t) is the concentration, and R(t) is the
evaporation curve. Thus the whole problem was reduced to solv-
ing equation (17), and showing that its solution R(t) has the
property R(t) < 0,

If now we would proceed in the same manner in the con-
densation case, it would still follow that equation (17) has
a (unique) solution R(t), but that R > 0, Because of this





















I









- 9 -



theorem [2] -3, although actually true, could not be proven to
be a consequence of (17 )•

The difficulty is circumvented as follows. Let R(t) be
the solution of



(19)



P - i • P -
u r (p ,t) + | pu (p ,t) = 0, p(0) = 1.



Explicitly, this equation is



{



oo = -2a
i l



\



r z(p,0) ,

y](T)dT * — — n(z(p,0))
1 2Vt '



-00



(1 9 ), j +f P^lf^ Z (p,cr) n(z(p,cr))dcr







- a



o /-Z(p,0)

PP J



-00



*

Tj(T)di; + pVt (z(p,o)



V - pf* P( a) p(^ n( z (p,a))da

r J VT^o- '

p(0) = 1

(wherever p appears without argument, the argument is meant
to be t ) •

We assert:

Theorem 1 Equation (19) has a unique solution R(t),

Theorem 2 , u R (r,t) = for r < R(t), t > 0.

Main theorem , c(r,t) = — u (r,t) for r > R(t) satisfies
all conditions (1).

We omit the proof of theorem 1. It is observed that in
view of theorem 2,









.









-lo-
ur (r,t) =0 for r < R(t),
hence, as r approaches R(t) from the left,
u*(R~,t) = 0.

Thus R, after all, satisfies equation (17), and since this
equation, as is shown in Section IV, has a unique solution
R(t), R(t) = R(t), and therefore R(t) could be determined
by solving (17) instead of (19 )• Yet we were unable to prove
that the converse is implied. This is explained in more detail
in the Appendix where the proof of theorem 2. is established,
The proof of the main theorem is the same as in [2], p. 9»

IV. Determination of _ the c ondensation curve ; Theoretical
considerations .

We observed already that R(t) is the solution of equa-
tion (17), and we shall use this equation to study the behavior

of R(t) in preference to the defining equation (19). Equation

-i .

(19) differs from (17) by an extra term, -^ pu°(p~,t), which

will vanish if p is replaced by the solution R(t).
Explicitly, equation (17) is

f •
ff = G(p)

(20) 1

1 f(0) = 1

where






•



























.



























â– 



- 11 -



(21)



*z(o,0)

â– '-a?



\ ' n(T)dt + -i— ■ n(z(p,0))

' 2Vt /



G(p) = -2a

+ /* ^4^ z(p,cr) rj(z(p,o-))da.



It is the same equation as in the case of evaporation, yet
because of different sign of a, the method used in [2] to
produce the solution and its approximations does not apply here.

As is well known (and easily verified), the solution will

-1/2
have a derivative behaving as t ' near t = 0. This suggests

that we introduce x =yT as the independent variable. To

preserve the form of the second integral we also introduce

s = "/a as a new integration variable. Let y(x) = y(Vt")

= p(t). Equation (20) is then transformed into an equation

for y,



(22)

where
(23)



' yy' = H(y)
y(o) = l



H(y) = 2xG(p) = 2p



2x



z(y,0)



â–  y -oo



n(T)dT + r)(z(y,0))



2x f X y(|)y'(s) z(y>s) „ (B(y#a))dl



x - s



(2k)
(25)



z(y,s) =



y(x) - yjsj
p = - a > 0.



(r.o) = ^ *l - 1



2x



I












.









i



i












- 12 -

Our object is primarily to derive upper and lower bounds
for the solution. Having this in mind, it is convenient to
consider a transformation K(u,v) on couples of functions
(u(x),v(x)), defined by

z(u,0)

'-00



(26)



K(u,v) = 2(3






n(t)dt + ^(z(v,0))



+ 2x i X u ^ )u '^ } €(u f v) w(.5(u f v))di
x - s* I



where
(27)



€(u,v) =



" /

,7^2 2 J

?/x - s s



U( ff )u' (cr)



"vIST



dcr «



It is observed that if u = v, then, since £(u,u) = z(u,s),
K(u,u) = H(u). We shall now assume the existence of two dif-
ferentiable functions u Q >v having the properties:



(28)



u (0) = 1, v (0) = 1, < u u< < v v' for all x >
o ' o * oo-oo' -

o

u u» < K(u ,v )
0- v o' o

v v' > K(v ,u )

O O - 0*0



These functions will be constructed explicitly in the next
section. As to functions u,v, on which K(u,v) applies we
shall require that



(29)



u(0) =1, u u' ' o „ - - o o
o















â– 



:.' .






- 13 -

Such functions will be said to belong to class f\, •

T heorem 1 , Let u, u, v, v £ Jt, and let furthermore

uu 1 < uu', vv' > vv', for all x > 0,
Then

K(u,v) < K(u,v), x > 0.

Proof ; 1) z(u,0) < z(u, 0) since u < u,

pz{u,0) ,z(v,0)

hence / ^(1)6.1 J -co * "^-co f



2) z(v,0) > z(v,0) since v > v,



hence z (v,0) > z(v,0), since u > 0, v > 0;



con



sequent ly Vi(z(v,0)) < >i(z(v,0))



?wr - - (r)

furthermore ^L£i < vy < V2 ,

V(T) - V' j

hence < £(u,v) < €(u,v) < — / x " a < ~ . But zm(z)

"•V / 2 VXS "Y2 /

z 2 1

= — exp(-z ) is an increasing function of z for < z < — ;

yw " "V2

hence, £(u,v) n(^(u,v) ) < £(u, v)ci(£;(u,v) ) . Since also
u(s)u'(s) < u(s)u' (s), it follows that

f X u( p )u ' ( p } «(u,v)»(5(u f v))da < f * {r 2 s) " ( ^ ) 5(u,v)w(5(u,v)Jl8.

0x-s ' "Ox^-s /

The proof is completed on adding the three inequalities.









.



â– 












t



- 111. -

Theorem 2 8 Let the sequences u ,v be recursively de-
^ n n

fined by

u (0) = v (0) = 1
n n

u n u A = K(u n-l^ v n-l^ v n v A = K(v n-l' U n-l ) '



Then:






1) u is a monotonically increasing sequence and con-
verges to a function u,

2) v is a monotonically decreasing sequence and con-
verges to a function v,

3) uu' < vv' for all x > 0,
l\.) uu' = K(u,v), vv' = K(v,u),

Proof of this theorem is implied in [3]»

Theorem 3 , Let u, v, u, v £• % , and furthermore assume
that u - v for x < a, a > 0, Then there exist two numbers,
A > and < X < 1 such that for x < a + A,

|K(u,v) - K(u,v)| < hd l.u.b.|uu' - uu' |

yo
+ l.u.b | vv' - vv' |
0
This theorem asserts essentially that K(u,v) satisfies a
"Lipschitz Condition". We omit the proof which is tedious but
not difficult.

Corollary 1. u - v.









,


















- 15 -

Proof ; u(0) = v(0) = 1. Suppose that it has been estab-
lished that u = v for x < a, but u f v for x > a. But
then, in view of Theorems 2 and 3,

|uu' - vv' | = |K(u,v) -K(v,u)|
0


Hence,



loUeba I'UU^ *• VV ' | < X l,U,t, |uu' - VV ' | ,

0


*v >v _ rJ



and since X < 1, uu ! = vv' for < x < a+A, or u = v
for < x < a+£ Since a is any number, it follows that
u = v for all x.

Corollary 2 . Equation (22) has a unique solution y and
y = u = v.

Proof : From corollary 1 and theorem 2, u satisfies

uu' = K(u,u) = H(y) e

The uniqueness of y is demonstrated similarly as
Corollary 1, on observing that H(y) satisfies, by theorem 3,
a Lipschitz Condition.

Thus the existence arid construction of the solution of
equation (22) for the condensation curve is established.
While we never doubted that this solution exists, the practical
importance of the above consideration lies in that the sequences
u and v defined in theorem 2 form ever improving lower and
upper bounds for the solution. Still there is a difficulty



- 16 -

in that while the first iterates are chosen as simple express-
ions, the successive iterates are usually difficult to compute.
This is remedied by the following

Theorem l\. Let u ,v € J\, 9 such that

u u' < u u < K(u ,v ) < K(v ,u ) < v 7' < v v' .
oo- oo- o' o - o* o - oo- oo

Then

u u' < u u' < K(u ,v ) < K(v ,u" ) < vj' < v v' •
oo- oo- o* o - o' o - oo- oo

Proof ; In view of the premises and theorem 1,

K(u o ,v Q ) < K(u o ,v o )

K(v Q ,u o ) > K(7 ,u o )

K(u ,v ) < K(v ,u ),
v o* o - v o* o '

completing the proof.

Thus, in view of theorem \\. t the functions u , v satisfy

all the conditions imposed on u and v , and hence may be

used to start a new iteration sequence u ,v , which will have

^ n n

the same properties as the sequence u n > v n °f theorem 2. Of
course, rather than to construct such a sequence, we shall re-
apply theorem Ij., to get improved lower and upper bound of a
certain form. Such a procedure will not, of course, lead to
an exact result, but, as we shall see, the improvement may be
considerable.









!



-





















.






y






â– 



- 17 -

V. Determination of co n densation curve : Calculation of
Bounds o

In view of the form of K(u,v), eq. (26), it is reasonable

2 2
to conjecture the existence of bounds u » v which are quad-
ratic in x. We thus assume that u > v can ^ e chosen as

/ u u»= ax + b u (0) = 1
(30) { ° ° °

v v' = k(ax + b) v (°) = !•

Our object is to establish this conjecture, i.e. to show that
with appropriate values for a, b, and k, conditions (28) are
satisfied, and to determine the best such values. For the
latter purpose we use the theorem l± of the preceding section.
Let us first study functions of the form



w = ]/l + px + 2qx , p, q > 0, p > q .
We have:

z(w,0) = ^~-— , z' = -^(xw' - w + 1); (xw» - w + 1)' = xw"

(ww')' = ww" + w' 2 = ww" + -talL = p.

w

Hence

t p P 2 2 2

w J w" = pw - (ww' ) = p(l + px + 2qx) - (px + q) = p - q •

Thus, if p >Vq > w " > °> w ' is increasing, xw' - w + 1 is
increasing and thus positive, so that z' is positive, and so
z is increasing. We therefore have




































•'






■ •



- 1-8 -



| = z(w(0),0) < z < z(w(oo),0) =^
q = w ' ( ) < w ' < w ' ( oo ) = V? •



Applying this to u and v which have the same form as w,



we get



!<«C V o) <^?



kb



-T 5 z ( v n »°) <



- 2
Vka



b <



u u'
o o _



2
ax + b



v



° }l + k(ax 2 +2bx)



v' .—

T £ k - f k



(3D



kb <



v v'

o o



k ( ax + b )



u



° VT



= ku» < kVa
o - v



+ ax + 2bx



- £
s



V



bJx - s „ # \ l->/a, /x -

2VFTT 2 ^VV ^ 2)k/x~+

1/ x + s - ^ v o* o - 2 Kx + s



kb
2



We assume that



(32) b <]/§ , kY£


The first of these is imposed by the condition p >Vq (and
it is anticipated that this will be met in our case), while
the second is required by one of the conditions (28).

We now have, using the definition of K(u,v), and (31)



- 19 -



K(u s v ) > 2px(l + erf(§)) + ^ exp(- 7^)

00-


+ ~1 X & -T ± \ C|)^=|)axp(-(|y|=f) 2 )


lA'o x^ - s



= L(a, b, k, x)



(33))"



K



. 2

(v ,u Q ) < 2(3x(l + erf(^)) + £§ exp(- ^-)

y-nr x - s y /



V



= U(a, b, k, x) t



If we require that



( U o U o - L(s '» b » k > X)



Oil-)



v o v o - U ( a * b ' k > x ^'



the conditions imposed on u > v > equations (28) will be
a fortiori satisfied. The integrals in L and U can be
evaluated explicitly and one gets,



L(a, b, k, x) = j 2(3(1 + erf(|))

I / _



(35)



+ a j^Jl Cl _ erf 2 (|)3exp(^) - erf(|))|:



+ f 41 exp(- ^) + b erf (|)l = ax + b



llA



TJ(a,b,k,xJ = 2p(l + erf (^p) )



1

+ JVS^S [1 . erf 2 ( k^ )]exp( ^a } _ erf( ^)j|



\ + f i& ( _ bi } +



£) + kb erf(^)| = ax + b .



*•(..-



'.









-






â– 












'


















! â– 



- 20 -



Hence, both L and U are linear in x, and inequalities (3I4.)
on functions reduce exactly to numerical inequalities,



(36)



a < a(a, b, k)

b < b ( a , b , k )

ka >a(a, b, k)

kb > b(a, b, k) .



It is desirable to solve the inequalities (36) in the best
possible way. Since there are only 3 constants to be deter-
mined and l\. inequalities to be satisfied, the best we can expect
would be 3 equalities and one inequality. Due to complexity of
expressions Involved this is Impossible to achieve explicitly.
But a computational procedure to achieve it is available in
view of theorem Ij. of the preceding section. We first have to
find any a , b , k satisfying (36), and then reiterate
according to the scheme



f



(37)



a n =

b =
n

k. =



£ (a n-l' b n-l' Vl 5
^n-l' b n-l» k n-l }



-1-



-liT/



n



= max (a^ a(a n-1 , b^, k^), b" b(a n _ x , b^, k^))



Theorem (Ij.) then implies that the sequences a , b are in-
creasing and convergent, and that the sequences k a . kb are
o dj - nnnn

decreasing and convergent. Actually, we do not need to adhere
rigidly to the scheme, but may at any step replace any one or two



of a , b , k by the same quantities with index lower by unity









...















.















â– â– _ .






- 21 -

to exploit the difference in rapidity with which these se-
quences may converge.

For the first step we choose,

a = 2 P
(38) \ b Q =f e-P

, k o = 2 *

The first two inequalities of (36) are then obviously satisfied.
Using the remaining two inequalities, we get the conditions



(2 >b. q \U q) b Q , 2)
(39) { ,

On observing that for all p >



(ko)



Vtt p(l - erf 2 (p))exp(+p 2 ) - erf(p)



= - 2 -f

Yfr



x



2 2

x - s



(pf-H W-p 2 r^f ) **



/



„ 2x f -Jx - s / 2 x -

-y5 4 p /~-p(-p ~

erf(p) < £E- ,
"Vir



- s\ ds



s/ 2 2
> X -s



Vtt j



do-<^



inequalities (39) are satisfied if
(1+1)



2 > 1 +2(1 + 2]/2) y^

2 > ep + mil >



that is, if












-






* •






- 22 -

(lj.1)' p < o0508.

Thus our considerations are restricted to this range of p.
Conditions (32) are satisfied if

i&e-P
2y2(3
hence certainly when p is restricted to the above range.
Actually, the range of p for which the method will \-iovk is
much wider, but we shall not attempt to extend our considerations
since the range considered is satisfactory for practical purposes,
and since for p = ,0$ the relative error in using bounds of
the form (30) is already .338 and it will increase with p.

One gets a simple improvement on (38) by iterating (38)
once and replacing a, by a , b, by b , and using inequalities
(lj.0). Thus one gets

(1+2) k x = 1 + 2(1 + 2I/2) ]/| = 1 + ^.32Vp ,

quoted in Section II. Further improvements must be obtained

numerically by successive application of (37 )• We consider

three cases, p = .00002, p = .02, P = .05, and in the first few

iterations replace at each step a,b by a.b, i.e.

n' n o o

iterate only on k with the anticipation that in (38) the ex-
pression for k is the least accurate. We get

o





















'






i •












- 23 -



J_


o00002


P 02


,05


k o


2


2


2


k l


1.0193


1.613


1.965


k
2

k 3


1.00571
1. 00563


lo319
1.214.2


1.722
1.566


\


1*00563


1.225


1.479


k <




1.2225


1.44S


k 6






1.431


k ?






1.4273



Using the improved values for k, we now continue iterations
on a, b and k simultaneously. In the case p = ,00002
there is no point in going on. In the other two cases we get



p = .02



n


a
n


b
n


k
n


5


.0k = a o


.0222 = b Q


1.2225


6


aOlj.07


.0226


1.200


7


.01+08


.0226


1.200



p = .05



n


a n


b
n


k
n


7


- 1 =a o


.0535 = b Q


1.1^27


8


.1024.73


.0560


1.362


9


.10506


.0562


1.348


10


.10508


.0562


1.333



We thus established that for all x,






I

i






!•-



- 2k -

(Lf-3 ) ax"+ b < yy' < k(ax + b).

On reintroducing the original variables t and R(t), (I4.3 )

1 2