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English Pages [212] Year 1949
experiments l
H. B. Mann analysis of variance
and analysis
of
variance designs
Professor of Mathematics Ohio State
University
AND DESIGN OF EXPERIMENTS
ANALYSIS by H.
book
This
Mann
B.
mathematically rigorous extensive discussion of an
a
is
important area in modern mathematics: design of experiments, or the analysis of variance and variance designs as statistical procedures. Emphasis is upon rigorous mathematical treatment, including proofs, formula derivation, and principles of statistical inference.
The
first six
chapters of this book cover the theory of the analysis full discussion of chi square distribution, F distribu
of variance, with
tion functions, analysis of variance in
oneway and rway
classifica
and
distribution of variance ratio when the null hypothesis is with inclusion of Tang's tables. Experimental design is treated in chapters on Latin squares and incomplete balanced block designs, Galois fields and orthogonal Latin squares, construction of incomtion,
false,
plete balanced block designs, factorial experiments,
designs,
blocks,
are considered
in
and randomized
and a
quasifactorial designs. Nonorthogonal data separate chapter, while analysis of covariance,
interblock estimates and variance are also considered.
volume has been directed toward three groups of readers: mature mathematicians who wish a knowledge of the subject; graduate or undergraduate classes; and practicing experimenters and statisticians. While treatment is clear, a knowledge of probaThis
bility calculus
This excellent
and matrix theory
work
is
will
be helpful to the reader.
one which every mathematician
in
any way
interested in the foundations of experimental design will find both useful and stimulating/' AMERICAN STATISTICAL JOURNAL. "Exposition is admirably clear throughout ... the book should prove
both
useful
and
stimulating,"
QUARTERLY OF APPLIED MATHE
MATICS. 14 pages of useful tables.
195pp.
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ANALYSIS AND DESIGN OF EXPERIMENTS
Other Dover Series Books in Mathematics & Physics theory of sets by
E. Kamke. Translated by Frederick Bagemihl, University of Rochester.
statistical mechanics by A. Khinchin. Translated by G. Gamow, George Washington University.
PROBLEM BOOK IN THE THEORY OF FUNCTIONS, Volume
I: Problems in the Elementary Theory of Functions by Konrad Knopp. Translated by Lipman Bers, Syracuse
University.
INTRODUCTION TO THE DIFFERENTIAL equations of physics by L. Hopf. Translated by Walter Nef,
University of Fribourg.
a concise history of mathematics by Dirk
J.
Struik, Massachusetts
Technology.
Institute of
ANALYSIS AND DESIGN OF EXPERIMENTS Analysis of Variance and Analysis of V ariance Designs
MANN
H. B. PROFESSOR OF MATHEMATICS, THE OHIO STATE UNIVERSITY
NEW YORK
DOVER PUBLICATIONS,
INC.
THE DOVER SERIES IN MATHEMATICS AND PHYSICS w. p eager, Consulting Editor
Copyright 1949
By Dover
Printed
Publications, Inc.
& Bound
in the
U.S.A.
To Harold
Hotelling
Contents CHAPTER
PAGE
Introduction
ix
Chisquare distribution and analysis of variance distribution
1
Matrices, quadratic forms and the multivariate normal distribution
6
Analysis of variance in a one
way
classification
....
Likelihood ratio tests and tests of linear hypotheses
.
22
.
.
47
.
.
76
construction of incomplete balanced block designs
.
107
Analysis of variance in an
The power
way
r
classification design
of analysis of variance tests
Latin squares and incomplete balanced block designs * Galois fields and orthogonal Latin squares
The
16
.
Nonorthogonal data
designs,
87
130
Factorial experiments
Randomized
61
139
randomized blocks and quasi
factorial designs
155
Analysis of covariance
169
Interblock estimates and interblock variance
171
Tables
181
vii
Introduction
The
idea to design experiments systematically and with a view to their statistical analysis was first promoted by R. A. Fisher in his well known book “The Design of Experiments”. Fisher also proposed the majority of the designs discussed in the present volume. Several designs of great importance, notably the quasifactorial designs and the incomplete balanced block designs, were discovered by F. Yates. R. A. Fisher’s book, however, as well as other publications by R. A. Fisher and F. Yates and their school are not written for mathematicians. Thus the main emphasis is placed on the explanation of the procedure with little or no attention being paid to a mathematical formulation of the assumptions and to the principles of statistical inference which lead from the assumption to the statistical method. Moreover, also in many other important papers on analysis of variance and design of experiments proofs
and derivations of formulae are barely sketched if not totally omitted. The present book tries to fill this gap and the main emphasis
ment
is
therefore given to a rigorous mathematical treat
of the subject.
In writing this volume the author had in mind a reader with a mathematical background of a student, who majors in mathematics and is in his senior year. References are given whenever the text exceeds this background.
The book is designed to serve three different purposes. First, was intended to enable a mature mathematician with no background in statistics to study the analysis of variance and it
analysis of variance designs within a reasonably short time. it is intended to serve as a text book for a graduate or advanced undergraduate course in the subject. Finally, it is hoped that this book will be studied by practical experimenters
Secondly,
and
statisticians who wish to study the mathematical methods used in the analysis of variance and in the construction of ix
* analysis of variance designs
and are
and able to expend
willing
the time and effort necessary for this purpose. thanks are due to the Iowa State College Press for their
My
kin d permission to include in this
book the
tables of the Fdis
Snedecor’s “Statistical Methods” and to the Department of Statistics, University of London, University College for their kind permission to republish P. C. Tang’s
tribution of G.
W.
power function of the analysis of variance test from the second volume of the “Statistical Research Memoirs”.
tables of the
I
am
indebted to Mr.
in reading the
knowledge
my
a very helpful
Ransom Whitney who has
manuscript and the proofs. indebtedness to Professor letter.
assisted
me
wish to acG. Cochran for
I also
W.
CHAPTER
I
Chisquare Distribution and Analysis of Variance Distribution In this chapter
certain fundamental concepts of the probability calculus are used. The reader who is not acquainted with these concepts should first acquire the necessary background by reading, for instance, Uspensky's, “Introduction to Mathematical Probability,” Chapter XII. Sec. 8, example 3, Chapter
XIII. Secs. 14 and 6, Chapter XV, Secs. 16. Let ai! x N be normally and independently distributed We wish to calculate •
•
•
,
,
variables with variances 1 and means 0. the distribution of the expression
=
2
X
The
+
x\
+
xl
joint distribution of x,
+
•••
xl
xN
is
exp [(x?
+
•
• •
,
,
.
given by the prob
ability density function,
P(x,
•
,
Hence the
•
•
,
=
xN)
2
X is
•
•
+
2
xn)/2\.
=
x\
+
•
•
•
+
0.
J
Otherwise the second equation would be a multiple of the first, contradicting the assumption of independence of the system (4.16).
The systems 53
a
V
adding over
b v g va
obtain
32 )
'Ziy*
The quantity Y a
is
(4 .
+
,
 Y a)Y a =
0.
a
variables g la
g pa
,
,
called the regression value of y a
E
(ya

33 )
then given by
is
E
= Z(Va Y a )y a a
a
(4 .
E yl  E n
=
a
.
a
Q =
.
•
•
,
,
g va
under
.
"aE
4 34)
~ Y a)Y a
(Va
a
Let now F* be the regression value of y a on g la the restrictions 4.6 and 4 7 Then similarly (
on the
.
The minimum Q a under the assumptions
r
yl

E Yf. a
Hence (4 .
35 )
Qr

Qa
=
E (Yl

Yf).
a
The
may (
4 7 ) are equivalent to stating that /3, be expressed by p — s parameters y, y„_,
restrictions
(
•
•
,
4 36) .
/
3
>y i
i
=
1 , •••
•
,
,
•
•
,
.
,P
i
Let
Ci
Then
• ,
•
•
,
Cp,
be the
maximum
likehhood estimates of y,
.
)
)
32
E
y * = Ec,
(4.37)
»..*.«
.
t
i
Multiplying the fth of the equations (4.30) by k tl c summing over t and l we obtain
Z
(4.38)
 Y a)Y* =
(y a
t
and
0.
a
Z° E(y« a
Since also
(2/«
—
=
Yt)Yt
=
y*)F*
we obtain
0
Z(2/«

y*)y;
a
 Z(y«  yjyj = a
o.
Hence
E
(
Ya 
Y*)
2
=
Z(^
 Y*)Y a
=
E y«

(4.39)
E
Y* 2
.
a
a
Therefore (4.40)
E (Y
 Qa =
Qr
a

2
y;)
a
and
F
(
Np «
EEa
(r. (
Va
 H)  y.:
2 2
(4.41)
=
at

y
s
Z« Z«
r*« 2
y°
 Z« y *  Z*
2
Testing a linear hypothesis means essentially testing the significance of the coefficients of a regression equation.
From
(4.33)
E a
and
(4.39) it follows also that
 Y a) = 2
(Va
E yl  E a
Yf
a
(4.42)

Z(Y
a
This result can easily be generalized to yield
Y*)
2 .
1
33
Theorem means
the
H
4.2: Let
•
l
•
H,
•
,
,
of the variables y a with
a sequence of hypotheses on
be
=
E(y a )
y a of the form.
V
Hi
—
Ha
l
Qiafii
y
*
H
Hi
:
t
&
H
•
2
•
&
•
£
H .i & t
a*,0,
=
0,
St
^
i
k
=
S(_j 4" 1)
** j
Y
each other. Let the hypothesis
1‘ ’
H,
£ yi
p
are independent of
obtained under
ya
then
z
=
(2/„

nT + £ (nu
+
n 
+
nr

2>
a
a
a
(4.43) •••
£ a
Theorem
j
H,
be the regression value of
such that the linear restrictions imposed by
l)
(
i".’)
2
+
£
(
a
w
.
very useful in reducing the labor involved in computing sums of squares of deviations from a regression 4.2
is
value.
We now turn our We write
attention to the solution of the equation
(4.30).
£g
(4.44)
ia g ia
=
aa
£y
,
be
4.2: Let g
—
a g ia
=
at
.
a
a
Lemma
(
+
+
p3 x
to be tested
" is
Xa
(?3
,
& =
,
,
,
where
,
„ = f
N
a
=
3a
.
The first step squares. Then if cl, 0.
/3 2 03 by least c can be computed from the g ia square estimate of 0 3
to estimate 0,
3
bar is measured at different In terms of theorems 4.1 and 4.3
steel
3
3 b3
TTq.
,
and
b3
is
is
then
=
cc
3 ,
the least
,
43
F
has the
distribution with
and
1
N —
3 degrees of freedom
respectively.
Example 3. We shall consider again the one way classification problem treated in Chapter 3. We assumed that we had taken a sample of n&’s from the first classification, n2 from the second and so on. Denoting by x„ the jth measurement in the ith class we have to consider the following linear hypothesis. Assumption E{xu)
The
=
for
/i
the regression value of x, f on Hi
is
7„ =
=

4.1
E*Ei
2 i
riiXi. 2
(
x ‘>)
,
•••
l,*,
—
(*«
2
m)
nt).
which
4.1 that
— nx  Ei «,x 2 2
.
the likelihood ratio statistic for testing our hypothesis. Example 4. We shall now treat the problem of a 2
classification.
As an example suppose that
different races receive s different diets
,
of all observations. It
and equation
E i
j
E* E>
mean x
=
way
rs pigs from r
such that exactly one
44 pig of the •
•
•
t'th
race
=
(t
The purpose
s.
,
• •
1,
•
receives the jth diet j
r)
,
experiment
of the
two
is
fold.
We
=
1,
want
with respect to the weight gains and at the same time we should like to know if the different diets differ in their ability to produce weight gains.
to see
if
the pig races
differ
Our observations can be arranged
where
x,, is
%11
)
% rl
j
*
*
*
*
*
into a matrix
j
%la
i
%r$j
*
the weight gain of the pig from the tth race which
receives the jth diet.
We
assume now that the weight gain is produced by two and feed, both of which act independently of
factors, race
each other. Moreover we shall assume that the x
and n
Mi
is
Mi
M,
a constant independent of
= X2
Mi
=
0,
the “effect” of the i
and j.
~
—
—
2
To find Q a we have to minimize X2. MiMi m) (x a subject to the restriction of (4.82). The conditions of theorem 4.4 are
however obviously
the restrictions. Thus
mates
of Mi
,
M>
>
if
/
m.i
m
v
.
,
may therefore ignore
,
m
are the least square esti
Zii
=
x
m.,
M we have
™ = ;o rrii
and we
satisfied
ZX i
>
j
i
£
Zo
rn
=
x,.
—
X,
= ^
Yj
z.i
 m=
x.,

X.
=
45 Thus the
F
regression value
Ya =
(4.83)
We now
Xi.
given by
is
( ,
+ x., —
x.
apply theorem 4.2 and consider the sequence of
hypotheses
(4.84)
Hi
:
Mo
=
H
2
:
Hi
&
H
3
:
Hi &
=
M 0,
H & Mi 2
seen that the regression values are
It is easily
YtY
=
Y\f
=
as.,
3>
=
x.
—
*•/
(4.85)
F‘
Hence by theorem Qa
+
Mi
Mi
=
X
=
S
(*
3
x)
—
r
»'
j
For testing
H
2
:
D
Q  Qa = r
*
,
Similarly for testing
Q 
(4.88)
r
^
(x.,
—
3
x)
—
rsx
3 .
i
=
Mo
0 we have

(Fi)’
F
3
3> )
=
s
£
(as,.

3
x)
.
*
i
H
2
:
/z.,
=
Q„
r
=
£
0 we obtain (x.,

3
x)
.
i
We
can further simplify
and applying theorem
p **

(r

(4.86), (4.87), (4.88)
4.1
l)(s
I
we

by means
of (3.1)
find that
1)
i
(4.89)
— —
rsx r
3
^2
x. {
+ rsx
46 and
r
#
(r

l)(s

1)
(4.90) r
— have both the
F
statistics for testing
The
—
distribution
—
— rsx — r yix!, +
2
2
x./
and are the likehhood
the hypotheses
degrees of freedom are r
E Z s
1
H
2
and
:
/*,.
(r
—
=
0,
1) (s
H'2
—
:
2
rsx
ratio
n.,
1) for
= 0. F2
;
and (r — 1) (s — 1) for F2 Problem 1. Find the proper statistic to test H l2 n = n 2 and H[ 2 )ui = M 2 in example 4. Hint; apply the corollary to theorem 4.3.
s
1
.
:
:
t
.
.
CHAPTER
V
Analysis of Variance in an rway Classification Design
Let us again consider example 4
of chapter 4.
We
had
rs
• r; j = 1, • • • , s). The observations (i = 1, could be arranged in classes in two ways and Xu was the value observed in the z'th class of the first and in the jth class of the
quantities x.,
•
•
,
second classification. This idea can be generalized and we shall in this chapter consider rway classification designs and their analysis for any r. For practical reasons r will be limited to at most 4 or 5;
however, a general treatment of rway classification designs is just as easy as the treatment of special cases and we shall give it here in all generality. To give an example of a 3way classification suppose that we have 10 weather stations. The mean rainfall was recorded 10 stations every month in 5 successive years. Every observation is then characterized by 3 numbers, the number of the weather station, the month, and the year in which the observation was made. Thus the observations may be denoted
by these
5), 12; a 3 = 1, 10; a 2 — 1, by xaia , a , (ai = 1, where a! is the number of the weather station, a 2 the month, and o3 the year of observation. We may for instance want to know whether rainfall was •
•
•
•
,
•
•
•
,
•
•
,
different in different locations or in different years. Differences
between different months are certain to be present. These simple questions do not however exhaust the information in which we might be interested. It is of interest to know also whether the combination of a certain location with a certain month has any bearing on the amount of rainfall, or whether rainfall was unusually large in July of a particular year. Accordingly
we
mean rainfall in one particular station durmonth and in one particular year as being month and year as well as of the the interaction of month and year, month and station, conceive the
ing one particular
made up effect of
of the effects of station,
47
r
48 year and station and finally one effect due to the interaction of month, year and station. Thus
F(X 0l0l0j )
&2
)
U3)
“f* ju(l,
2j
m( 2, 3; a 2
,
a 3)
+
3; a,
2, 3, d\
+ +
(5 1 )
a*( 1;
,
ai)
+
u 3)
—
/x(l,
m( 2; a 2 )
+
CLi
,
Ct 2)
,
a 3)
+
m(3; a 3 )
#1 ,
where m(1j 2, 3; ai
a2
,
u3 )
,
01
3,
/*(!> 2,
cii
u2
,
>
^
“*
m(1j 2, 3;
1
)
/*(*1
;
I2
®>,
)
=
®i a )
)
®i,
»
“•>
= X)
For instance of station
d
t
u2
,
,
month number
5.
The assumption can
"f"
/i(l,
also be written as •Pataaaa
m(1> 2, 3, d\
+ +
(5 2 )
d2
m(2, 3;
+
m( 2; a 2 )
d2
,
U 3)
,
+
d3 )
,
2j d\
m( 1, 3;
*
,
Uia),
• ,
(5.3)
* y
t
>
*

®»*)
=
0,
* *
*
)
ti)y
:
'
49 where the second summation a = 0 and 1 .., n(u mation over all combinations
£
r
with
a it
kp
atl
;
•
•
,
•
,
o tJ )
(5.6)
—
*(*1
’ >
•
•
j
ia+i
a,,
;
•
•
,
•
,
a,„ +1)
a
Z(ir
T
70
x(b i
We ob x
compute now • ,
• ,
in the last bi
fl6
T ).
sum by
.
’
»
' '
>
by
The term
a:(6i
• • ,
for every choice k x
Out
;
a bl
•
•
,
•
,
in 5.6 the coefficient of
of the
are for fixed 0 exactly
a
+
1
% ,
,
by
k2
;
a»,
•
•
,
numbers
•
d by ).
xfa •
*,
• ,
,
by
;
a tr ) occurs which contains
•
•
,
,
,
kp •
,
•
•
,
i a+1
there
*
51 such choices. Hence the coefficient of x(b x
•
•
,
•
,
by
ab
;
,
,
a by ) becomes

s
(
+
 1) "’(“ = (i )“ +i
r

(i)'(“
r )
7 )
j
1 (l)** *.
= This proves
7
r
+
5.5.
We show next that the solutions in 5.5 satisfy the restrictions = x(l; a — x. We have by 5.6 is clear for A(l; a
in 5.3. This
x
A(l,
=
•
•
•
x(l,
,
a
•
•
+
•
1; «i
+
a
,
(5.6a)
x)
)
•
•
•
,
•
1; at
a«+i)
,
•
,
•
,
a„ +1 )
•
•
a
E E
—
A(ki
,
,
kp
;
•
Oi, >
•
•
•
j
0 !,•••, o+l
5.6a over a x and applying mathematical induction
Summing we obtain 52 A(l,
• •
•
x(2,
+
a
,
•
•
•
1; ai
+
a
,
•
1;
EE
•
,
•
a a+ 1)
,
a2
•
a a+ 0
• ,
,
a
fl0 2,
by
• •
,
a(*,
,
kp
;
a*.
a+1
ik now shows that the A(i x • a„) are the unique solutions of the minimum problem. form quadratic minimize the wish to Suppose we
The o
•
5.4.
•
( ,
following argument
•
•
;
,
,
•
•
,
,
Q — ) f
°i
) i
*
*
*
) “•«
x(il
1
• J
*
* /
ia
t
' '
toil
)
*
)
Qia)
L
—
^2
00 *!,•••,*«
m(^i
‘ >
‘
i
®*i
' >
T
’ '
>
“
52 under the restrictions 5.3 and no further restrictions. The solution to 5.4 which as we have shown satisfy the restrictions are then1 the uniquely determined values for which Q' = 0. Hence if we would write out the least square equations including the terms resulting from the Lagrange operators we should
a k,
• >
We
•
A{k
solutions for the
kf
• k
,
,
;
ake) since Q' can take only one minimum value.
>
apply
now Theorem
H
— Hi &
If
same
get the
still
•
yf* *
2
&
•
•
4.2 to our sequence of hypotheses.
&
•
,
y
j
*
* 
j
dik)
* ,
•
In
£
£o«..» ~
A fa
times.
n:i.... b
r
bi
each
•
£
•
,
•••
ik
;
,
^
i z
a0
*
1,2,
*
,
=
• • •
•
•
•
lo
)
‘
!
a i„)"
)
r
£
Q=
52
52
a
!**>*«»
#*(*
i
>
r
(5.9)
•••
•[A(ti
,
i
,
'
Mtfl
' '
»
)
‘
fa
»
‘
‘
l
l
Besides testing hypotheses concerning sets of interactions m(»i
• • ,
•
i
t«
;
«i
•
•
,
•
a„) for
,
•
•
all
,a a we
•
,
may
also wish
to test hypotheses which concern individual interactions. In
such cases certain sets of interactions n(i ia Oi aa) will be assumed to be equal to 0 for all a„ We shall refer to such interactions as interactions of type I. Other interactions n(il ax a>„ ) will be unknown ia for all di a„ These we shall call interactions of type II. In one such set of interactions, however, we may wish to test hypotheses concerning individual values of the set and we shall call those interactions of type III. Equation 5.9 shows that for finding Q a and Q, we must put n(i, a, a a) = 0 for interactions of type I and n(i t ia a a„) = ^(*1 »••»*'«» «i »**•
•
52 at
•
• ’
52
[AO'.
• • ,
•
,
jt
;
o.
• ,
•
•
,
,
at)
a*

2
mO’i
• • »
'
I
jk
;
a.
• • ,
*
,
a*)]
54 with respect to mO'i
'
ju(ii
'
•
•
•
j
"
)
,
at
jk
'
)
I
!
jk
I
• • ,
Ol
•
under the
a*)
,
)
>
>
restrictions
0
®t)
'
ffl«
I
a«
and certain other
imposed by assumption and
restrictions
hypothesis.
As an example consider a three way classification and assume all 2nd order interactions are 0. We wish to test the all interactions between the first and second
that
hypothesis that
The assumption then
classification are 0.
m( 1>
2, 3;
tti
a2
,
(® i
=
The hypothesis
=
is /z(l,
2;
a
x
=
a, 3
,
lj
=
a2 )
>
=
(a t
0,
£3 )
•
1,
... < 2 ). The number of linear restrictions h a 2 = 1, imposed by the hypothesis is (ty — 1)((ii

2
iiX
.
at
Suppose that 5.11 *
* *
°t‘i
is
23
[^(^1
*
*
7
*
k.
ik
1
)
From
5.10
&
*
*
ix
^»*)]
*
7
we have
7
°t* *
*
3E]
(5.12)
r
1.
and consider the Galois
The order
Every element of this Galois field + akx k k < s where a0 a, •
•
—
a divisor of x
of this Galois field
of the
is
• • ,
form a0
is
a,x
+
mod
p.
+
a k are residues
100
+ax +
(a 0
(8.21)
•
•
x
•
+ a xY k
=
=
since x*’
x(J(x), p)
a0
+
bound for 4he order in our Galois Theorem 8.3 since s > r.
Lemma p
8.4:
)
x \9 2
•
'
>
ffi
>
j
x
gi
02
)
)•
The system thus constructed the element
(0,
1,
•
• •
,
1)
is not a field since, for instance, has no inverse in multiplication.
However, the postulates IIV for addition and IIII for multiplication and postulate V are fulfilled. All the “points” which have no 0 among their coordinates possess inverses. Let o,
=
i,
fl4°,
be the marks of G.F.(p”‘), then
•••
if
,
r
g$“'
•
)
k
+
K
v lv i
contained in P.G.(m, p“)
of P.G.(fc, p")
becomes rq 4
+D n
v
)
(P
p
+
m>
n )
,n
p
,n
)p
>
s
0.
contained in
is
+
+ pQ
(p"+
)
•
o/ a P.G.(m,
p")
B. Every line
•
•
(p
P.G.(s,
)’ s
contained in
is
n
r
p
one obtains in particular
•
•
2
(p
P.G.(s,
p
+ Pm •••(?!»+•+ + (p”) + p«) /or m>s> ")
•
”
•
•
n )
•
•
p)
•
•
1.
Every P.G.(s, pn) contains with every pair of points also the whole line joining them. Thus every pair of points is contained in X different P.G.(s, p").
We may now (s,
p
identify the points with varieties and the P.G. with blocks. Then we have the following theorem.
)
n
Theorem 9.1: The P.G.(s, p ) contained in a P.G.(w, p") form a balanced incomplete block design with the parameters (9.6)
h

(1
+ P" + (1 +
'
•
•
'
=
b(s,
v
=
1
+ p* +
•
•
k
—
1
+ p" +
•
•
+
•
+
•
^ ,n
p
'
+ +
fa'"
(p' 1 ’"
)
n
m, p
),
•
•
+ p mn = +
p’
n
=
v{m, p
n ),
k(s, p”),
•
•
•
+ pQ
p’^p’"
.
111
+
(p" r
n
)
+
(p
=
+ pmn + p”)

(P’
•
•
+ Pml
•
 1)B + P*> (p (
•
• •
r(s, to, p"),
=
if s
1
2
(p
+ +
"
•
•
•
•
•
1
=
.
+ pm + p“) ")
2n
(p
We
+
n •
•
•
,n •
•
•
m,pn)
X(s,
+
(P
•
•
•
•
(p'" 1,n
•
>
s
if
•
+
m
+
P l n
P'")p'
1
next consider the points in P.G. (to, p") common to a  1, p”) and a given P.G.(s, p") not contained which is not contained x
given P.G. (to
Let p be a point in the P.G.(s, p”) n in the P.G. (to  1, p ). Let q pendent points in the P.G.(m — 1, p”). 1 linearly independent points are
in
it.
•
x
•
,
,
m +
of
P.G.(m, p")
of the
is
XiPi
Now
let Pi
,
•
,
m
linearly inde
Then
pi
,
•
•
,
•
,
qm
form
+ •
p2
qm be
and hence every point
^2