BASIC METHODS AND DIMENSIONS IN MECHANICS

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R001200240001-7
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 405
• Document Creation Date: December 27, 2016
• Document Release Date: December 5, 2012
• Sequence Number: 1
• Content Type: REPORT

Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
R
STAT
Next 1 Page(s) In Document Denied
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
11,SICTflOrx3 AND D11,:,. 4 1014,:;
IN ISCligI CS
- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001
Declassified in Part-Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
Preface to the First Sdition
In physics and in engineering, in experiments and in practical calcsulationb !It
is constantly necessary to consider various facts connected with the similarity of
phenomena and with the dimensions of observ d quantities. The ?7onstruct.i.on of lir-
planes, ships, dams, and many other cornplex technical structures is based on pre-
liminary general investigations, amonr, which the testing of models plays an impor-
tant role. In the theory of similarities and dimnsions the conditions which shoul,!
be observed in experinents with models are established and the characteristic and
appropriate parameters which identify the basic effects and therates
of processes
are evolved. Together with this ombininE the the theory
ties and dimensions with a general qualitative analysis of the mechanism of physical
phenomena in a number of c5e5 can serve as a fruitful theoretical method of re-
search.
We encounter the problems involved in theories of dimensions and modelling in
our very first studies of physics in school and in research work in the initial
stages of setting up new undertakings Moreover, these theories are distingui.hed
by their extreme simplicity and their elementary nature. Despite this, the theorems
concerning the similarity of phenomena have only become widely disseminated and con-
Sciouily utilised cooperatively recently, i.e., for example, in hydromechanics only
Within the last 30440 years.
It is common knowledge that the presentation of these theories in telt-hooks
e and in actual teaching in the higher educational establishments usually suffers from
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
77.,r
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
.r
many defic (motes. rhos(' questions are touched upea only ensvAlly And in passing.
The beet concepts, evon the concepts or dimensional and dimensionless quantitiee,
the problem of the number of fundamental units of measurement, etc., are not explain-
ed in a clear manner. Moreover, befuddled and intuitive representations of the con-
tent of the concept of dimensions often serve as the starting point for the emergence
of viewpoints in which the formulae for dimensions are ascribed a certain mystic or
peculiarly secret.significance. In certain instances such confusion has led to para-
doxes which serve as an object of perplexity. We will analyse in detail one example
of such a mieunderstanding in connection with nelels? conclue one core:.erning the
heat exchange of a body in a liquid stream. In preeenting the theory of sinilari-
ti 3 reletionehips and mathematical devices which are not essentially connected
with this theory are frequently introduced. It is d sirable to de eve tne construc-
tions of the theory or dimensions and irellar4t-Aes as is generally the case with
any theory, with the aid of methods and basic premises appropriate to the essence of
the theory. Such a construction makes it possible to cl.early outline the limits and.
possibilities of the theory. This is especially necessary in the case of the theory
of dimensions and similarities since one often encounters extreme opinions: on the
concerning the onrnpotince of this theory and on the other, its triviality.
one hand
Neither opinion can be considered correct.
It should be noted however, that most realistic and useful results can be ob-
tained by combining the theorems of the
of general Phieice, which, in
theory of dimensions with the propositions
itself, yields interesting conclusions. Therefore, in
-illustrate various applications more completely we will consider a whole
eerie* of mechanical Preola= and examples of
various types.
the eombination of dimensional nethods
other qualitative mechanical and mathematical theorems.
This his likewise inpelled us to concern ourselves with the problems of the
turbulent movements of a liquid in more detail. In the theory of turbulence, methods
sitdierity are the 'basic working theoretical methods
since, in this field we
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
?
Declassified in Part - Sanitized Copy Xpproved for Release 2012/12/05 ? CIA-RDP81-01043R001200240001-7
still do not have d system of equations which make it possible to reduce the
mechanical problem to mathematical problem. In the section on turbulent movements
of 4 ]liquid new results are presented which eupplement and clarify several problems
.11 JL _ _
invo4,veu in vale valoOry 01 Talrouloncei
In addition to the examples of the use of the methods of dimensions and similar-
ities we have attempted to shed sOMO light on the resolution of a number of mechan-
ical problems wtich are very important in engineering and certain of which are new
and have as yet only be slightl. veloped.
with general theorcw concerning the nature of various mechanical
relationships and also in connection with certain independent values, we have dwelt
in somewhat more detail on an examination of the basic equations of mechanics which
express Newton's second law. The point of view which we will present is therefore
not raVls however, it is considerably different from the treatment of this basic prob-
lem in mechanics as it appears in certain widely used text-books on theoretical
, mechanics.
The number of known applications of the theory of dimensions and similarities
in mechanics is very great and we have not touchod'upon many of them. The author
hopes that the book will give the reader an idea of the procedures and possibilities
of these methods and will help in the analysis of new problems and in the formula-
tion and execution of new experiments.
No special preparation is required for reading the greater part of the book.
In order to understand the material presented in the second half of the book, it is
necessary to have a general knowledge of hydromechanics.
Moscow, 1943
This book is an extended and revised version of the book Methods of the Theory
..-rsf Dimensions and the Theory of Si imlitude in Mechanics published in 1944.
_Declassified inPart -Sanitized Copy 'Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
?.?
7.71,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
In the eeeond edition addition to cer aln. corrections and minor improve-
41011%.3, supOlOmmontal information hae been introduced in which the theorena of the
theory of dimeteions are utilized to determine a series of precise derivations in the
theory of a wave on the surface of a.heavy ideal liquid, in the theory of the move-
ment of a viscous fluid, and in the theory of one-dimensional n nsteady states of
motion of a gas. By an analogoue method it is possible to seek and establish the
mechanical characteristics of motion in other problems in matbervticaI physics, for
in the.theory of plane-parallel and spatial steady-state motions of 4 gas,
pie
in the theory of the dispersion of turbulent jets, etc.
In Chapter IV, solutions of a number of problems concerning one-dimensional
nonsteady-state motions of a gas.. which art of signific nt practical interest, are
gi
G.M.Bam-ielikovichwhom I epreas ry sincere gratitude was of great assls-
tance to ne in writing Chapter IV.
I am also grateful to I.O.Betharev under mhqs supervision the calculations in
the problem concerning a powerful explosion were carried out.
Moscow,January 1951
L.Sedov
Preface to the Third Edition
Recently, theorems and methods which nake use of the property of invariability
of mathematical and physical laws in selecting units of measurement and physical
scales for the characteristics of the phenomena being utilized have penetrated much
!ICTO extensively into scientific investigations.
The practical and theoretical value and strength of these methods is being re-
cognized more end more among scientific workers in opposition to the opinion which
widespread until pat recently that the theory of similarities and dimensions
was or secondary importance.
certain analogy can be spoken of between the theory of dimensions and similar-
STAT
iv
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Ities and the geotric theory of the invariants connected with the Lraufsforaw.L.;
coordinates the fundamental theory in contemporary mathemAtics and physics*
After the publication of the first edition of thin book many new applications
of the thelry of similarities and dimensions to the most diverse problems in physics,
the mechanics of complex media, to certain problems of a mathematical nature connect-
ed with the utilization of the group theory for seeking solutions of differential
equations*; and to statistical problems selecting and rejecting merchandise -J1:1 in-
dustrial products**.
This edition contains several corre:tions and additions designei to
or emphasis to the fundamental ideas inherent in the theory of 5thlaritie5 and
41imension5. Trne, for example, this has been done in the course of the rils
The definition of the dynamic and, in general,
0:reat-
of the proof of the
physical similarity of
new definition is still
less from the practical point of view it embraces all the essential peculiari-
Moreover, it can be conveniently and directly
theorem.
sion
the
phenomena has been presented in somewhat more detail. This
not generally used in the presentation of similarity prob-
ties of physically similar processes.
utilised and evidently, completely satisfies all the requirements of a number of
applications.
Sections 8-12 in Chapter IV and the entire Chapter V have been added. The addi-
ions in Chapter IV are devoted to problem involving explosions and the extinction
of shock waves and to certain theorems in the general theory of one-dimensional gas
motions* The now Chapter V is concerned with the applications of the theory of one -
*In.this connection, we take note of the recently published book: Gairkhoff,
HYdrodrnamits* A Study in Logic, Fact and Similitude, 1950* There is a Russian
translation edited by 16I.Gurovich: G.Birkgof, Gidrodinamika, I.L.? 1954.
-,*S4141 $ .Drobot and 11,6Warmus, Dimensional Analysis in Sampling Inspection of Merchan-
dise. Rozprawy Matematyczne, V oWarezawa 1954.
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81
-01043R001200240001-7
STAT
4 Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,
,
t
i
dimensional, nonato ? -state gam motions an0 the methods of dimensional analysis to
certain astrophysical problems.
It has now become clear that the basic problems of the internal structure of
stars and problems connected with explaining tha grandiose and surprising phenomena
observed in variable stars are closely related to the study of the problems of gas
dynamics. The theory, as it is presented here, gives new rational formulations of
the problems and presents exact solutions for equations representing adiabatic gas
motions and gas eqtilibrium equations which take radiation effects into account. In
certain cases, appropriate idealized instances of gas motion or egoilibrium can he
considered as schematic processes which serve as models of gas dynamic effects occur-
ring on stars. They cn provide a basis for gaining an insight into the possible
mechanisms of explosions of stars star pulsations, the internal structure of stars;
of the effect of various physical factors connected with the release and adsorbton
energy inside stars, and the role of variable density; of the influence of gravity;
and of possible movements o, asioned by the absence of an initial equal distribution
? of pressures, etc.
? The theory developed in the additions to Chapter IV and in Chapter V are to a
? significant degree entirely new. The proposed formulations and solutions of the
problems of gas dynamics can be considered as illustrative examples of applications
of the methods of dimensional analysis to astronomy and as a reserve of model, sim-
ple, ideal motions which can ,be drawn upon and utilized in the investigation of cos-
mogony problems. Some of these results were obtained by myself and a number of my
. young students in connection with work done at a seminar on hydromechanics at Moscow
;tate University during the academic year 1952-53.
N.S4?But-7aviendS.I.Sidorkin assisted in the preparation of Sections 8 and 9 of
Fter IT; VAArasiltrov and 14411:0LidoT, Section 10 Part 1 Chapter IV; N.S.Burnova?
SactiOA 1; Chapter IV; and X4,!Yavorskaya, Section 6 Chapter ?V.
'ilc.praas MY ainaia gratitude toall of them.
Afters': wrilmou 10CL
STATL.Sedov
1111111111111111111111111111111=1 in Part -Sanitized Copy Approved for Release 2012/12/05:- CIA-RDP81-01043R00120
7
,
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Chapter I
GENERAL MORN! OF DIMENSIONS FOR VARIOUS )UANTITIES
1. Introduction
A whole series of conceptel for example, energy, speeds stress, etc., which
characterize observable phenomena and can be assigned and defined with the aid of
numbers has been introduced into the study of mechanical phenomena.
All questions concerning motion and equilibrium are formulated as problems of
defining certain functions and zumerical values for quantities which characterize
the phenomenon. Moreover, in solving such problems natural laws and various geomet-
rical relationehipe are represented by functional equations, usually, differential
equations.
In purely theoretical investigations, these equations are useful in establish-
tag the general qualitative properties of motions and in actually calculating the
functional connections being sought by means of various mathematical operations.
limgaver, it is not always possible to carry out an investigation in the field of
mechanics by meansof mathematical reasoning and calculations. In numerous instances
?in solving a problem in mechanics insurmountable mathematical difficulties are en-
countered. Very frequently, ve do not have a general mathematical statement of the
?problem since the mechanical pra0Amtbeing investigated is so complex that there is
? still no eatisfactorT. mathematical representation of it and no equations for the
motions involved. W.4, encountor such a situation in solving many very important
? gpsstions in the fields of aeromechanics and hydromochanics, in problems concerning
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
'
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
the study of stability and detonations in various atructures, ete. In these cases
the principal rae is, played by experimental methods of investigation which afford
the possibility of determining the most rudimentary? 'empirical facts. Generally, any
inveetigatioe of natural phenomena begins with the determination fef rudimentary era-
pirical facts on the baeis of which the lawe mere:Leg the phenomenon being observed
.can be fOreulated and recorded in the form of eertain mathematical relationships.
In order to set up end conduct experiments , the restelts of welch can be -utilized
to establish laws and car, be applied to ises in which the experiment can not be
carried, out directly, it is ne aviary to examine the eee,ent ial na ...ure of the pr ob e
being studied and to make a general qual itat ice analysis, Furthermore,
,vee ,:oreula-
tion of experiments whose refoulte will be represented by numerical corthinations which
characterize the aspects of the phencmenon being inevetigated, nay be carried out
only on the basis of a preliminary theoretical analyte?. In setting up experiments
arid generally in practice it is extremely important to sel,ect the correct dimension-
less parameters. Ilse number of them should be minimal end the parameters
'upon should reflect the basic effects as closely as Ipossibee.
The possibility of such a preliminary qualitative-theoretical analysis and the
selection of a 'system of specific dimensionless weneters is afforded by the theory
of dimensions and similarities. It can be applied to the observation of extremely
complex phenol/one and significantly facilitate the performance of exper'irsentv.;. Fur-,
theienore, writing up and carrying out experiments without taking the quest ion of
dimensions and similarities into account is unthinkable at the present time. S OM -
tine et the theory of dissonsions is the only possible theoretical method in the ini-
tial stages of the stUdy of certain complex phenomena. ?However, the possibilities
?for the 'fise of this thod should not be overestinated. The results which can be
obtae4 using the theory of dimensione are I.imited and in new cases trivial. In
0014jUrtetticortk with this, the rather widespread idea that the theory of dimensions can
not itfil4 any illeportant results at all is conigAtelY untrue. The combination of the
STAT
decided
?
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
? ?,
-
!4,4*
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
t similarities with. preesone , ned from experimen a or by mathe- .i.al
means from equations or =Lion can lead to rather essential resul s. Usu-
ally', the theory of dimensions and stisdiarities is of great use both in theory and in
practice. All result!, obtained with the aid of this theory are arrived at very s
;p
ply, in an elementary manner, and with hardly any difficulty. Nevertheless, despite
impleness and elementariness the application of the methods of the theory of
dimensions and similaritie5 to new problems, requires that h investigator have ap-
preciable experience with and fundamental understanding of the phenome-,being
? studied.
With the aId of the theory of dIirensions it is possible especially val-
uable #oncluions in observ tions of those phenomena which d pend on a largecum,ber
of paramettrs, but certain of these parameters in known instances
are unessential. Later on we will illustrate such cases with examplos. The methods
of th theory of dimensions and similarities play an especially important role in
ou4.,
Dimensical and DiMensionlest titis
4
Z,vAe
he numerical values of which depend
Ati
epted scales, ide., on
?5ystma. of units of me surement, are called imensional or concrete quantities.
?Quant1tes, the numerical values of which do not depend on accepted units of meas-
urement, are called dimensionless or abstract quantities.
Length, time, power, ener
lt of force, etc., are examples of dimensional
tontines es relationship between two lengths the relationship of the
guar, of length to s, surface,the relationship of energr to the moment of force,
etc., are examples of dimensionless quantities.
the subdivision of quantities into dimensional and dimensionless is
tu a certain extent, a matter of convvntion; thus, for exale, we have Just called
an angle * 41$111 quantity. ut it is a. known fact that angles can be meas-
inradians of a, right angle, i?*., in various
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
STAT
1,1
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
n. angecen he considered aaa entity having dimensions,
ratio of the aro of the circumference subtending it to the
this defines the unit of measurement of the angle, the
now measure angles only in
in geometrioally siedler figure
rresponding lengths are not identical
to select different distancee for the basic length.
considered dimensional quantity, the dimension of
owever, fixing the
inconvenient. This
, corresponding angles
therefore in different
Acceleration is e
11 is length divided b
for of gravity 1 which is
be considered a constant qeantity (9.81
fixed unit of asreVEflt rorac
d as the relationship of s quantity
In many problems the acceleration of
on of a falling body in a
titY of acceleration of the f?
rl the 'Immo al value of
to another*
But, the a
h will not be changed by transition from
,sequently, an overload is a dimension-
an overload can be considered a dimensional
of measurement is taken am accelera-
geantite, namely as loratiOnwhen the uni
tim equal to the acceleration of the fore of gravity.
that an acceleration which is net equal to ttle atneleration of the force of gravity
can be used as the unit of measurement of the overload*
other hand* qu&ntities which are abstract (dimensionless) in the gener-
be expressed with the aid of various numbers.
iS can be expressed not only in the form of an
tionship of two le
lout but also to percents or by other means*
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
r which the unit
aSUren*nt we will
in theoretical
ferent nits of as
lows that certain
and dimensionless quantities are relative con-
t of unite of measurement. Then the qeantities.
surement are identical in all accepted systems of units of
all dimensionless. the quantities for which in experiments or
ions it is actually or potentially permissible to use dif-
urement'we will, call dimensional ? Using this definition it fol-
uantities can be considered in one instance dimensional, and in
above and will encounter others
umits of measure n
quantities will be expressed in a specified manner through the unit
Sent of the basic quantities* The
es will be called basic or pr1ary and all others, derived or secondary.
In practice has Proven euffcient to establish units of measurement for
quantities are connected to one another by definite relation-
basic and certain
are establiahed for thms. units of measurement for all remain-
of measure-
units of measurement taken for the basic quanti-
s on the cone ret conditions of the given
oblige and for different Problems It is efficacious to use
different quantities as the basic units. Thus, in physical
convenient to take units of leg400) time. arid as
units of measurement of
44.
IrritZ?,460.T.04.w44oLfr.A.Q
s as basic units and in engineer-
units of lertgth, time, and forte. But, units of measurement such as units of
viscosity, or density can also be used as basic units.
At th present ti physical and engineering systems of units of measurements
used xKet extensively. In the physical system the centimeter, gram-mass, and
the CGS lint system) are accepted as
ering system, the meter, kilogram-.
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240 -
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
equal to
); and time, one second have been deteredned by. the experimental method on the
basis of specific relationehipes The length of a standard made of a platinum-iridium
whichaUoy is kept in the French Palace of Measures and Weights is accepted as one
meter; the mass of a standard of the same metal kept in the s" p1e is accepted
46 ne kilogram. A second is accepted as part of A.n averaEm solar day.
24
the basic units of mess rement ha been stablished? the units of
d accel-
measurementf other mechani al quan iti a A for example force, en
oration c., can be obtained autortical1y from their definition .
The exprssicn of derived wtits of measurent5 through b.si it.s of asre-
nt i called ans ion. A 1insion I destribed symbolically by a formula i.
which ol 04 he unit of length is the 1tter L; Iszrs,M; and ' tme r (the
01 of the unit of force in the enireeringls 1). For ex ple, the dimen-
sion of a square 1 speed or force in the physical ytei, ,...a.. 0
MLarid in
4.
the engineeringe 2(4.
In the fututo desiate the dinsion of yquant.?ty will use the
*)1 [a]*
.or exazip1e, for the dUirs1an of orce F in t.e physical system we will trite:
NIL
nt to another* For exam*e? in asuring the ce1.ration of the force of gravity
? timstors and **cods wehave g 9e1 ciiVs9c2. If it is necessary to go from
se units of measurement to kilograms and hours, to convert the indicated numerical
lues for the acceleration of th. force of grivity mUst use the relationsh
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIX-RDP81-01043R001200240001-7
NIZEIMISSW
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
" so 981
or 98 II 404 .
ew systems of units of easurennt the unit of length is
a times, the unit of mesa times, and the unit of time 0 times less than the old
units, the numerical value of the physical quantity which has the limens ion 1.1 -
lia Olfne is increased in the new system aleyn times.
The number of basic units of measurement need not ob1iatOriY be three. In-
stead of tbrea umber of tas c units ctn he used. Thus, for example, 1-7
imans or eXeriJent5 it can be established tha
q n ies length time 734t3 3
case, flewtonts equation takes the form
the units of measurement for the rour
jependent of oneanother. In this
c a constant having
With such a selec ion of ba5ic units four arguments will generally enter into
he formula of the j5jnnion of the mechanical quantities. The coefficient c in the
les' constant similar to the acceleration of the force
equation given 'Above is
f gravitt g or the gravitational constant Y in the universal gravity law
shmlit
are he masses of two mstsrial Points and /el the distance between
rical U. of be coefficient c will depend on the selection of the
STAT
Declassified
in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
r:11
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
(consequently, c will have
n. all systems of measurements) an equal or unequal unit,
dimension of force will be defined by mass length, and
be defined simply in relation to the
h, and time.
by introducing upplementary physical ccms ants we can select by
axperirentLtl means
independent of one another, units of meas rement for n quantities
hould introduce n, 3, dimensional physical constants.
he de ivative quantities will ontain in general n
In stu mechanical phenomena sufficient to in
dependent basic unit5 of measurement' .0 for length
One can suffice with these units also in the study of
phenoia. Fru physics it is well known that the dimension
ical cuantities can be expressed by Ls N4 and T. For example
intro
three in-
force), and t
hernial and oven electrical
of thermal and elec-
the amount of heat
he temperature have the dimension of mechanical energy. However, in practice,
thermodynamics and as dynaitics units
t
of a temperature which are independent of the units of measurement of
of measurement for the
Degrees centigrade serve as units of measure-
0riee, for measuring the amount of heat. These units
can be established by experimental means independent of the units of
onversion of meohanical energ7 into heat occurs,
necessary to take into consideration', two additional physical dimensional
ent the miihinioil equivalent of heat
it yell A44
.he other,
either the heat capacity coefficient c n13 degrees or the se
I?TAT
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
11
Pint
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
C on t
or the Bo3.timann constant w 107?1016 ors/40
02 degrees/ /f we measure the amount of heat or the temperature in
moshani al units then the mechanical equivalent of heat and Boltzmannts constant
will be introduced into the formul- as absolute dinnsionless constants and will be
analogous to transference numbers in converting for ex
into kilogram-meters, etc.
It is not difficult to see that the number of basic
meters into feet, ergs
s of reasureent can.
also be less than three. Indeed, all forces can bc tred with the force of r7c,v-
ity, although this is inconvenient and unnatural in those instances where the force
of gravity plays no part. In the physical system of units the force in general
is detormined h7 the equation
d the force of gravity by the e uation
whCre y
vitational constant which has the dimension
Similar to the manner in witch the dimensional constant for the echancal equivalent
of beat can be substituted We the dimensionless constant when measuring the amount
of heat in mechanical units
ational constant can be considered an absolute
,Oimensionlees constant. In this w47 the dimension of mass in relation to L and T
an be d:
r.43r*
in 'this oese the change in the unit of mass is defined completely by
11?).'?0
??.?
ISOPeOnrelSen ? '.:. .f!?*!.'???? ?
"d tine* VmS, considering the
,:qt:41,,,pp4,qpnstant absolute dimensionless constant, we will have two in
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
STAT
0??
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
den4ent units of u ?ureunt in all.
The number or independent unite of measurement ean be reduced to one if we take
one more dimensional physical constant, for example, the coefficient of the kthema-
tic of water v or tha speed of light in a vacuum co as an absolute dimen-
sionless conetant.
Finally, we can consider all physical quantities as dimensionless if we take
appropriate physical constants for absolute dimensionless constants. In this case,
the pa 1rilty of using different eysteme of units of measurement is excluded. One
single system of units of measurement based on the selected physical constants, i.e.,
on the gravitational constant, the speed of light, or the coefficient of the viscos-
ity of water, is obtained,the values of rnch are taken As absolute universal con-
stants.
In scieflce a tendency to introduce sUch a system unite can be observed. This
due to the
that it can be used to establish units of measurement which can
not be lost, like the standards for the meter and the kilo quantities which
are essentially MANN quantities not connected with the basic phenomena of
According to the ortginal idea of the commission of the French Academy of Science
which was engaged in establishing the ntric systemof mea5 , one meter was
defined as ----I-404 the length of the Parisian meridian, and 1 kg as the weight
4x 10'
of a cubic decimeter of distilled water at 4?C. Naturally, the measurement of the
length of the ridian and the prepereca401 of standards for the meter and the kilo-
gram were carried out with a certain degree of inaccuracy which later accurate
meaauz'ementa showed to be greeter than the permissible percentage of error. Since
in increasing the cisinn of the determination of the length of the meridian or
literth. weight of. a of pure water new deviations could be anticipated, in order
0 11
constant changing of the standards for t
,
ter and the (continued)
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
STAT
?.x
,1441,
,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
Ilke intr>uti
ther syt
n of 411,c1
0 ayitom of tnt of meAsurtmentS wh h excluAes
uremvmt5. io?eqpilalont?to the toMplote elimination of tho
neept of he dimen on, In the tingle universal ytem of units of measurement
the numerical values of all quantitative 4harcter3tic are defined by thiLr physi-
?al quantitys
e
tain
hips, A similar un
n., .t)le use of ident_
al meas tem
represent a definite convenience for p
linkz in the standardization
owever, fo
ersAl sinae system of units of nea,
f1.4
of ,111u1nt nrtine
iN ical purposes, beirii crtai,n cf the
f methods of measurement.
many phenomena, such
stant, the 3pee,.4 of light in a
cuum, t,
of water, are coaplete y I teri1. her
of measurement connected with the laws of
friction in water, o
? artifical
rious branch
?unite of
ative aigni
observed.
In meahan
ties,but it
oma other ph?v
ure and would b
convenie
al con ta
cc
the gravitatio al con-
of the ktic rot'
un-
f6 si er A. system of 'uni?
ravity, the propai7ati 4 of light, or Is
processes in many ca
,sacti e. On the other hand, in
of physics it would be convenient and practical to use systems of
ent with different basicunite corresponding to the nature and the
xcena being
ieance of the
concepts watering into th
it is convenient to use force, length and tifle as basic quanti -
re convenient .o use other units for forte and length in engineer-
ing mechanics than in celestial mechanics.
OUS to
trupproproommorrogairoomorr.
Ott intled
In electrical engineering it is more ad
the tnhofaurrent, resistance length and time (the amp-
14 the standards as the basic
101
S 0
with
to solopt the quantities of prepared prototypes
s of istasurement ard to refrain from connecting
loSbh ot a
or the weight of a liter
AAA
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
STAT
;IQ
Declassified in Part -Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
ohms centimeter, nd
Moreover, for the concrete inveetigation of individual
pecial classes of phe-
nomena, be numerical values of quantitative chara teristics are frequently advanta-
geously expressed in relation to given quantities which ar most haracteri tic of
c'ses, these characteristic
the particular problems being considered.
basic quantities can be different.
unit
the ?ormi1a of a Dimen
of the units of measu ement of derivative -7uantities to ti?.e.
The re
nship
In different
of measurement of the basic quantities
formula is called the formuL
definition and
??an? be represented by a formula. 717.5
of the dimension and it can be onidered as a concise
haracteristic of the vhysi::al nature of
It is possible to speak of
unit5 of measurement.
dimension for a
be expressed in various
In variou
tity c
dime
7
derived value.
ion. only in rel ti n to a definite syst m
ems of
its of measurements the fort.ula
iffere
t number of arguments and
of CS units of mea
es ave, the ro
of
?n
of
surement the for-
.1,Lo., to, oloftlOttliA
VI IditC FOWcw1.,
cndittn the ratio between two
he selection of a eca
area in s
dimension is defined by the following physical
numerical values of any derived quantity should not
for the basic units of measurement. For example,
re meters or square centimeters the ratio between
re eaters will be just the same as the ratio of these
For basic quantities, this condition is a
unit of measurement and is itself satisfied.
simplicity we will assume
take-11100' Joon, derived ; for
cue, atuuttito? is geometric and therefore depends wily on the length'
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R00120
Declassified in Part-Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
certain distances. Let us deligmt by yl that value of the
quantity y which corresponds to the values of the arguments xlf, ?
4
numerical value of y and also yt depends on the unit of measurement for the distances
Xlp Xviso, xn. Let us reduce this unit or the scale of the distances ?
Then, according to the condition formulated above, we should 1 v:
the ra I
d be idertia1 for ay valm where the :,ale of the
es of the derived geometric Tzan
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
integrating, we find:
Since when a ai w have p 1, thn C quen,
cinclusi is justir ional quantity which de
ral basic rantitjes *if we cnly change the scale. It is not ifficlt to f,1 0
then fpil1 have
This proves that the torrulaa of a dimens on of physical quantities should urve the
licsns of Powered ,..
? 5 ? On the Second LAV' of Newton
---------7------------------
chanical or in general PhYsi al Phsnomena w will introdee
em of oncePts, es, quantities wtdotl characterize various aspects of
rocossee being stu4ied (we simply characteristics), arid secondly,
the t of units ofmeasurement with the aid of which the numerical values of
1.491 which have been introduced can be determined.
of relationships exist between the characteristicsnomenoni
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
"P
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
10 'le
,Certtx of these rilationehi 1st onlY for con?rete eye ems and for isolated p,..? ?0,4.4 4?
the ro7on.'.,,ent of atioiirt 01:1013 t..1-:';at s1.1.1.pk(fIrtv.
?
us 3e.e.4 a.
r' Cfl the tA1ronntrYlti
14 9 0
determine the potential the
form to these ,conditions.
,
ppJtenI
net, be able t,
,
,(Alk Or} 4 4.1
1, .?)
co:nAtionsbi
give .,).tc!.re.to
ITTOM linftarity vroblen it follow:, that it is su,...-.1cient to
& the? ;a stir,
w(z) depends linear (the constant a eaa be complex:).
?
fa] tyr
tieas,xmotions rade we see that a
veters can be represented b7 the tabull,l,ion
t.o=p2e'e vsFer of dete.,L,....r.ing para-
-
Let us now cure
wz=a:/,(:,I tot-
_ ,
the exponents i an0,..0 have been selected eo that x'is an abstract quantity. Since
?
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
t I , n theoryiit e
1:0V/6
IEEE
real run.
)
-
f:(3_
het on unn a liquid and Orl free
*thich 1-a iiniu1;r:
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
ct on 1,
nece ,
1.11"41-g',1,1V1 1'7*
n the. acteristicfncrLjon
int,,eg,:r7te the ,lifferential eiT?tati,on
the mbst general c
liclu?ici .aroarri've t.
if w'at that
regular., then fr.
s
ity 4 -onet
st.7.nt
, ,
mot ionti 1ich hal!
nu-e ,?? c!..,...;:ert?e?....1
we come the ? probleln
t.he 77.1f,:gmrl.e.:It of tc
.coL,D:,?1111t1'm
t 2:
r)?-
?
d "ht,".1"tre r t I t i
for
tJo nctJc Y (.A.)
,riti.e 3 0:1 .t11.6 ee o
fl
evizr,.4
4, ,4
4 15a arbirar4,? }.1.
?,,
The bac3i
eyw
1 7
t1,
41*0
- , to.i5e?e. ? that
41.
,ertaf.,,n wae notion.
te cre ee .be .7a.1v..e
e solu ion equat?o (13.13) i expres?i in terms th (.-or luent
c1.4.!on8. 7equation*
y which7 0 dif.,
yi ky
gener.i salt), on of e 3) takes the r
0'
Table 0 ion with i?oriao andI'VeSoCTI, 1949.
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
wh re
n
?
ok
'r",-?
-
t
4. 4 *4 in,
cho,ra,7
on f
V.:..C) .!`?,
1. .0 is a whole number.,.
responds to a simple source,
the above considered solutions for
an be, obtained .from the solution
I
7
4 ? 23 )
different,iation with respect to ti.me:
^
) CI
which cor-
(L.21.1
Hence it is clear that these solutions correspond to tine dipoles whose orIer
is determined by the number k. In these cases the character of the variation of the
- coefficient of correlation f (r/ V;T) is represented in Fig.22.
I
It is not difficult to see that fors., .4 the parameter A possessesa fini
value different from zero.
On the basis or the eq. (4.17) we can writeik se follows:
I ? 0
(v1)4/2
b(gr dr --(t) di,.
2 toe
0
NM 0 00040.4,
LLn loqo. m nllifinnhhikenr_ DAN SSSR.
a er
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Th qua1it ixdi , es that the eon 'cy in tiNe of 0 oo is incompatible
with the inequality. - z, From the expansion (4.21) LL is evident that for 0er
he case o ,,h, ent r ...on ,.,ici were carri out for all possible values of the
Reynolds number cona_ z re1 th correctness. .
of a emala
..., axis en t.
vers w go erninE the distribution of velocities,which 4 irthpen1nt of the
Tknelds number (see Fig. 23). aready a3 e
ini rical fortu
presentat on of the results of the experiments for the quanti.?
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
.23. perimental Confirmation of ?Fhe Jniversa1 Lav uover
Distribution of Velocities Close to he Ad.s of the
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Irt
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
"
an a function of the
and Vikuradze
11-4
* Ales
dz .th 'works .oT Stantor? 'itritsch
authors note4 tho existenco of t
indica 1 universal de-
pendence, whieb turnel out to be4h
.,-re t in the central p rt of smooth and0
1 3 inct pendent of the rouhr,oss, 1 spite of the fact that thc resist , aml
urax
also the ratio --- - ,tc. eel 'Aen1 ?4poL the Ileyn Lls ntu:,ber a. , )or the
v,4
16.
preserite4.
,:?'roil. an arla
-sis of ,
?, ?tiGn thich gives the 1A1
4o3ri ectio of the se of ,
,
w '1,fc
vallle
the
?i?,e
.;?
case is
or the distriuut1or of the vra17?t3iocit1e3 wc obtn tAe roll ula
Stanton, ft'oo lo- Soo. (A)0 vol. 850 1911.
Fritsch INI`lr 1928 and Aaohener Abhand., ;Tin, 1928a
zel Prcceedins of the Third InterratioraL Congress
a
. _
Declassified in Part: Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
- -
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
tho cas, of In,rdn'nr
m'4.114)d. theoro In fAc:t1 in tho c.a$e or .f,4:11:lar .motion
zve uni 0,t4m, anl re. tillneari ? thorefore the pm7
erty.. of irtia .1.1Pt, be :non. , ard c nsequently the stributon or the YO-
.4"0,0 wr, ntity P SirIce t' etors are.
a
ral case t biet " otor ar th
- _ _.,..
exists . dnar ier, ,,,r,en n,. a . v., 1.,,,,:i
a ..,4, 4 - ,' ''. the .,,.. --;" a ik.,.
hat in the 'thborh ol o the wall, .
r trbu n '.p.(11)
ul , 4 . -I. . ...a.n.h i.et...eni . with :,
x?
t3r1rent5. But in erents a .tp With fLc ru there is aJ,:le:: also ,he
r
ir irtvetiyatioi of the ?flstribuUo2i
of th velocturbu-
in the case
_
?Le T. xrstior a great role has be n played by the powr empiricai foru1as of the
es
e A a4 r ar constants. Thco con8tants can be ietorriine
r ot ;e trexent, of the 'el it ributioa or i ectly with the aistan.ce of
an ontai c1eter1natior of the lawgv, i re is ance of the "pee For
-
Declassified in Part - Sanitized -Copy 'Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
err,z of th
the pipe.
Th coerfi
..of the ?latt 1 methol .14e extro , th. ro;Aotan.ce of p1
ttte velocity .4.1.:otr ?butioa with 1,oliteet t',.,1:?4 OC
of re
tanc ,)
is eradno.l. by. the equ.a
*. or .
over
, for t?,,he orr
1r or
e ? .4 .A -and do .not, 1..erclrid uponwe oh&i .
ett*i h;
or.-th coa 1cet a rorru1a ofe
e velocit
Declassified in Pari- Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
pt ,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
? .fx'or;?;) 41,71..ch we ,1 terran ?.the.
attin whi0A we rILd. on the 'basis a the
OA
pressi
`24:1
Co the: 64,S? (5.12), ob t;;; e bt,ltwee.
stants 4111 the c.; nstA?nts a,
"rhe ? lad, a
for the
-
wer law
.e.a13 to the zo a1,1el aro- se7er
rix
? Thetat xt g.24) show that the hest cojjejc1ence withper-
i s +.airles,1 forth owl r-
4?p, * o a
( 5 16
ead
A to coo , eien n tiii b
experiments .
Declassified
in Part -Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
1-1
2._
0
'
??-'.
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
A p
The eqs. (5.9) and (5.1b) lose e,rreatness in the iTilmeliate neic,hborhood of
:the t1ls.,? where n , 0. 4e4r. the walla there is a laldinew f3r T(7.1)
It we assume that the larvinar layer borders on the turbo, ent flow and 1de require
:that the velocities of the partielea of the r1/411 at the bol,,tri,dar7: of the lnar
layerpass arer continuolAsIr into turbulent Aistribution of velo7;ities, which are
10
7,0
. The Di tribu
? '? ? ..`;
? ?
t^ ? ' ?
?.
J?
,
r'10'
3
?-????. ??-?,?; rL a-,,-ter
4 ?
rl Of Vel. (414 4 03
deterl:'inA4 according to t'..,he eqs. (%L5) and (5.16), then th akes it possible to
determine the thickness of the laminar la-fer either from the followin equatio:c.
or from trie equation
5,75 ig
?
1=8,71'17.
Theso1ution' of these sgtlations th both cases gives for a value that is close to
,
aPProxi.t..Mte3..,Y., Setting n ,,,, 12, we find the formula for the thickness or the
t
c,
.., .
,
laminar 1,47er to l'clet the follo!il1n4;
.24 7
,
1*0
0
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
??;.
Yir.f
-
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
10 employ_nr; the -orm A f n'
it c& be conciliAte that the thi
.pek.ison w1t.h the ra1i ot the' pi.pe.. A,
I mean value of the tr
included ?
4,005 ,01
Isills? we Obtain:
ayer i 2] in cm,-
considera ions of Prand,t1 and, K ::ar con-
of us de
o_s the velcol , of th
01 LI.?
the voluI-,e of the
tion of theTurbu1exLVoti
r 0,71in1rit%41 Fipe
Declassified
in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
th Z&L c 0d." , And:114r t,. to we hiave 0 and. we Alen. have Poi:lel o.
t ease t,herv...16 obtainod? from
the 5 ) ? tile ? parho1jc law .ppvern,j,
t.he ve-
1 t Urihut In the woe or turbu.1, en trot or la, t, 0 1,7 1oar the .ror
0 t here in i..arti..n?ar _Ivor. 4
0 and a pproxi ely
CONZ0-
-?
qe of whi oh eq, .1111 adsLi fo I ow relat on
A c7 Z1, he
,we urn our ttor t 0. In this casco t,-he cii,x-ec.tif.7n of trrt to the cco1-c)sponii1714,7.
eurve?4.7,-; Is siteur$1 for COI"Iti t ant, velocity c turrIts.: count er-
,
,
lb ts ession
p illereafAes as.
bNitiou," ,goirvt,teer/hre''?of lde,r.tsly1=t
1...,t1:a,i11,..i,itifr'''ens91.z.t,Yxf, or) auatts'ellt:,,airt'zi*?)n', if,erbat et:0?6111:1:
'phockAs tho
re'- cedes from e ee,R (),r'Yt near' over a .1, ,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
i4; ronolvf faitileath
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
ver.1 t
or 104
? =, 4 r
.4, ,4
i
Dar;etr
own fmcUou3
of t
4 on for d .var-
u a 0 of gas
TIO 10 /On Of
ptn. . .
tar:
zTo? +4,7
? . e
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized.CopyApproyedfor.Release..2013/,1.2/?9:,
717.4 V7" " AP
.? .
, ? ? ? ? ?
,
?
'19.4 In in th ropoItiort t.he ttu'buirtnorI
t.
? . ? , ,, ? ? , . ? , ? . ? .
r . r
towards :a t, ra:m1 ng ..14'ave in 1,4;111, c h t ,i xod nt of. time are sit uat c?Td
. .
.arl. I, t e 0. .ancl wUch ' n ti rti wave oray..' by
? ? . r r .
the ri.1.6r:e tic 'icto.13.1 e.o ttal.rioc4ty, 's o talc! y
qu', rIt 7.1:14 t uz
tatt c Lt:1:1:,%t le a 1.
A') ? , A,A, AA.... 0." 4 ,A.i./0,0. AA :4,'4 10" 0", 4; f
e orr esoj ethOtt Sp reasoning and re crried ovr. , f
or,,,, . ..,.,..1. t ,..,, ?.... ial, ..11.,,,t, . n , 1.,1. rr. , 1 , r., r, ,1,,,,. 5',.,.? ,,, .e,,, r. ,,,''" ',,,...''., Y,e,,, a., ' vv.... ',1. i4 ,?' rs-r*--, . ,'
. 1
.,
, ?
I
. ,
,
,
1 ,
, ?
t,'.'1....i.,i, .,-.,.1 cin,i, - t,,,,..,....L...r.aN, 1_ ..d7'; e' t.....r 't e, 71
, 0 Tx ,T.,..1n1,.; ,)??7 ,
, .
?
,
. .
(r
For
t c
,
414,e,? Etion p r 0 -a z
?,0 . 4, 40- A o
. ,
n the ?dir e ti On nf o ord r differ rola .1',
? 77?
. or. ir th 1re r in th(??,..,forr
. , .
. . .
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,'A'ic4v
4
c;?:4?? ridqf r
gAlsall ? ....tVVX.":*
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
101i.
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
t
' .011 (I!)
,
ar.id A.
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
11 p t a
?ion
? t..? ..q.
? km. f1?4'..ort o Lh. ,1).11.11.1). c- a
4.-)y
con.taflt,,
bai of
syrptoUc o
otiori cf th&3bohtn
be " weoJtam
and for
4,1
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-IRDP81-01043Ro61200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81?-0,1943R09,1,30,2740001:7,
)i 0 ``). t ? ,nt:(:. ' 1,
V0,1
.,; p a 1.7)
, ,r,cq.yv.z
to'
VA" S+,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
-7------
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
A10 Cii OlArin
ere in the ft a It on
t onf'
lea in trraf:,; Di7
vk
4 r
- 41 r
dir Lo , , :rrit.
t14 a collat.,
,.. t
4,v 1
n .. , ..3 4, i A mt i 1
4 .
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
f1 ic Ln; roluttor),
Ili ,,11 a .4.: fi t I .; a ppr..0 ,,I..:3 ,,,,,, we c',. 04 , in , e'l ?.' '. n. ,apq 1( ..,..i.rtf '
?
, f.-.,.?1 I val e to
Cro * 4, , ,.:1, 3 f,,..1,:v;?,.(. i. l?, ,,,,,,,A . ,. , ,'' h.
in o.:(7:,.(2,3 )nt t 1,, a,.1 . on
1 '11 i i
:.nci, (1.1 . n rv? ?? , e.:,';.,:5,Lo -?'..,::-:-. tl. -,3x .- '.. a e,. ' ) . 1 r, c.) .c; .t., ... QI-Ei
t 1., , s f: 11. t. 1, Or ,l'..' Or t..! l,4, c 1,-,,,,,,, ., 1 :,,,- .., n 1 '":i
..., , 311U
? , \
4 '.0,4' ' 8 I.) !;.,3 0 1 ... 1 ; i',.', nor 1r:,t! :;ri,"" t? '.:. ,, , I,
Ana1ogoi in th c'uo Of e
Declassified in Part-Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R601260240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
0(1.(1.1,12)) it r o110',.,;',t1 tip.1-1t,
,
41) (4)
A
,
p., . r-
t
?.,.
...,,t,
,
, ,r4.0 e . 11,7,4 .42 v
os-'," ; 44' 4 e
Za I 4i r2 (it7
r t 41.t..etrrat,iort
- 1
?
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
rflditifl3 at the jump ?i ,r,r4e3porle. 71
the for
e eb LflQ*
thg cbrctritic in theId towa
shoe . ay dijne or he chaacteri3tiC.
*,..- ..
,,4 Id4' al o d
".,..t ..tor %., ...
hod t abovec
- _
Declassified in Part - Sanitized Copy Approved for Refease 2012/12/05 : CIA-RDP81-01043R001200240001-7
fi y
j.$
, Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
t3k1
,ey1/4t on.
thomed.'
0 et :Lop t,
t 1 I
h1 th?n s ,tt a r
Ylte
?
d
bier.
oast ant
rite r s 0,4:11.12:
a.
col'iat
the ia
trt?
- 4,7
4, "1
t5,h
4' BO
kraerk
MNIIMINI1
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
41't (
I I
*V't 1) (
r or 5 el oiiin, motion
blem tf a trorrr OS 0 in tion 7
?.1
t ,0-1 ea
\ ?Le ? 1, J., roof/
Declassified in Part -Sanitized Copy Approved for Release 2012/12/05: CIA-RI5081-01043R001200240001-7
,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
1., :to 4,
i i ha, a) tl= if _ Li 1 -1_ it 4 1 t - *
' '" ' go '7 4. ? 1--; -r' f i iyAl 47 wl- 4 *
1,4; ()'i4
II ti ,i, Li dit . ? ..%
? OC'T "Sri gill /44, "t* ni (iii q
Pi
A..1.1t ? 1.1,?erini, no d
n 01 tat 0xi c
5
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
'v?1tigi`4.0.'"
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
e' loi ot. oquA, on6 ttio. ? .1o10:14n?? 0.
Onti
.-0'..1?c!))j, 4010 'S.:F.:1,011
06to 911 1,0t,6 thv
bt . re perltejd. rOrip
oro?th4,
1tror?,Hto obtain....aII or ( ?e :
_
_
.,
r c nT,-, ,A 4 f r -4 obtain '"1 ? 0 t e 1 ations , n--
o ,1.. 1 a41 ..k? '4.f .1.
0 tr fr the genwa1o1ut1on of the 7 of o da.nary eqtatio under
,
,
,
co dei.at on,'' itch t obtained Cron ystt'rna
dons .8 t o a . that noar tile cmtcr ,..,,i.
....?- ...1 V.4 r?, naIy 1rt t oUo,
STAT
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
A
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
trig form Ar
0 " kt40 Plt
t, 4 2P
fl
? 0
)%'111 (70 -f- T TtlX1" +
re )trct .rInt ? .d.6.,,per0.
tr -t?t VZ.1, .4. v. .1:';.e ? .1"'
t., In 9 .....ili ? .
? ? .:1 f . ?
...:(1i.298,2,..?
Q.',;31071?' 991
.0.... t)8571, 1cI33Q,..... . ...?..Q.0.0,1,3,.
0;3(3.55 .,.... - ,...0 . ,it... .91979.
,
1537 d1X6' ? ? J
1,4296t9 '1,12138kii
, 7 9f3(4 dal.* V**
[ 1 dal'. ?
6 3759 + Vita
21,8045 +
5, 932V*4
9t 1837 diX1*5-t- ? ? 41
and the c at1ions fro
Declassified in Part - Sanitized Copy Approved for Release 20'12/12/05 : CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,c?-7 77 0; '???? ? '
+ [6,1.537 + tilXst g
7,0358 1,960411?5 .1 his ?
2,5714 1 7208k**6
6,3759 W') fi
1,
t $
d7k.,1
maz
Aftirtitiet4, --01" +
?3,7474)--$?5
iovrt-
s 5680X6
+608 7 41?$'
e
kt
-n,
the pro
n mra a
a tjle,
0 uti 011`
1?.
raziner.
co
ue o v
110 .
? r
k1411,4.01;
=MIN
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
? ?
. , . .
611.arititfY???
Tho ? ?? ? .1" ?
?-?.re.,sult
? ? ? ? . ? . ?
?rt.k, 0,1 4
ift" 4
.
woo:0 r
r th'' 1)r,i. vatii7e$
6?\
(X ) and i
q jq *6 ? ., , 1 4 t . )(11,),",i,,W,1:111:,4 re'r''olf)Ptilers511 le,
\
".4'.' 4 rACt 0,,f7 Cotinten,wers:sur-e . III t?-tWe ,r11 ,3 , ,
".,',. .? the 4411? .,, louion ? .
-.? - ' Point' 1:-,P -
. ,
,--..-....._
ei4 h .
helptile. 1:-1.,f' this r 1
g1,10), :1'n' ?I'(')/44$"'ne' ;' 4thh t'I''''''l tItluati'arl'''' (::1':4*) fa' r'''
,.. , ?, . . . ..
, q 4,,,is.f,;(),ezibie to, ts,,,stbrate tile .c/14r1f3e vith.,,;.1,-1137..tge of t .;e:t' or the ch'a-
rstic or Eiri, raotior,i,"
? . '
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
? ?Declassified ?
in Part
- Sanitized Copy Approved
for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
't
' ? ? ' ? , . ?
wy.$00,7400..,
'? ? ,
':???"? '
? . . ? ? ? .? ? .., ? ? "?
t?-?11,?,.... ?
?
. ? ? , ?
????????? ? --????'''''? ?? ? " ? .?
? .
? - ?
,
79 ? t as a Punction
- 6 e- the dynex6?.c timeinsoconqs,..
,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
or tb 1..?Qh 16:0'4 40.011- r..,;;?.
fin 7, t 1.r;,(:),,t di ffi-
et0P4Wil
f'or small
I
C 41,
14 .1
he.A et1 ;
IOC
A I.
un
0 1 or el of 4 o n
ot--
-te
ns -Li c rkterPr .". 4A.6
1. 0 TtIct viues or h charac
nct
_
on atrnosphtric conditiQrn3e gin in corriapond
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
Se ton rtLT' oi vati
rt. "
1. II d :LI ??? 't,-,
-.,,I.).-u
4,r,, 1 4-, ,-,
, .rt. .u.re .11;
_ s ,r51.0 14,:vi.
t.i=4 Pi - ",f1 r i' .... * u, 1 t '
1,-0
4 t. 0
x
%a t ' 0
k e 110 t . 1 ti, t ;Lon.
:elms -'
ble ' . the,:. 4.;,..tot , 0 .?,, d 4 .'e of ,4.7 (24 Ina, . rori..$ ox
, .the n ''.4 . to.. re-.
witththe
0 :).,-, ?.,..,;. o ..,
,t a I.
r A wor.
centImes hal q.s i main source o ?x e ,
rind of m de
a
phenofla aunt lie the pxing ard so1ut iofl o
o gas which ho con dr
on e ov 4, are ab
paasing 'the
A per
or the inves11OTi
ruirer of drazrlc pr b-
asthoreicai itode1s
nebu-
u xi a
it is
c 0
lication to th p
' Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
P II
'. 'Ich 110 ' 1 i , e o .. co4...104,1 a t to 1,,, Ilikk 1 t!tld 0 ?
It 1. r
J. .5 .1,.,
AV ..
,,,, - , b ,0,1130:jkon or ,,,.e stars was earr I opt, aimp ..y: 1;:$y eye so
the stare or'th tuec,? vo, itvid,,, Ap?..pi?::(.0 tO 'be brigh ;11 stars of the sec-.
4 4 iy,,,44,44,40T t d to brii::51OTt-t n the etJ3"Tt5 or tho
('. so many
It A t lk. 3 ;0 and ro , on . this 1,tit e all the Stars were able- to oe arranth ged
ospond '1 t
evlee IA 10' their. apparent $htress.. Already in ancient
e a
,rAs the star rto 41uring observationsbr the.nako eye.
Th.bright .ar vg rl? refer I, ; x aTid t,`,` ail.,e0 r t.he P.Lrst
s ,arS
.40
whieb? i, 't t L:',..
1 ,dtude , t4.he ar 1.3 0 ba gnittA so on.
,
i .11a..:. '5 ntOint. vv.,. 1 414
X40 ;$i e r tbe . 4. oz , in cor, spo d -
obViO
ice with ts t ,, 0..pparent magnitude -e,:e ..? qa
:ters. :Vr071 the jrtiCat dGt.. , . o t 1 Ai,01,4 , that4-1-; t'rts: tilQr.rte-r 1? itrf-T:ethe
e thent t ;in
' -.)
a et
1,,h 4'0 I n 4
Ln r. jOUS CO IC 1:"5
A.? Anh 4, le Aw, e
.110
to . el tiv ,
to
olet
al
th
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
follows we Shan ut LiJ.ze th concept of oi.om?trjc, sta. 1.ar sulfp' tudo.
t is ObVious that the steliar magnitude in is letermined by the quantity of
11%.1111A4.414.11t
'
':.the'EsrthHfr(* 'the ..Star und r
flriton, Thi norg )por t ta. energy which i radiated
th star per unite ht i, to he i inojtv L of 4r 13 in ersely
, .0 from the, Ea r t b t the sta .
) o i n A. 11 ULtre )1. I
n the bas ,of he nropertjs o th human eY'ef- which '1,,,, determine,. by' the
WeberFchnr ii ha been r )1, I that iur1rg Aria tion o tte 3 t ell a r 1 ' gni tu es
T. , .,r.;-,
r Leh are de t ed by f,? t,,,, ( t,he ne c
,.., -
;s .,, , t,..es the ir itant
t 4r , o an .. 1 i the c r fk g"r: nci
geom.( i - i,
,...4. % it.,, vary in, a .,..z N . Ice i , the n'Al q?s
A
.) 7
0 a g rcmnt 4t this w thre , es shf 0 fol.rel t.lon
. he, ren v., e of
. ',
a ,
i equal o - . - hence i:. follo ws
11'Sir 1in otherwords, ,.
ne apparwit rajttie of t
essential:. e dIt.cepf
.ar to the
'thereforeto obtain the comparative ch.ra.cterjstjcs..
of re weof b o
ce theconcent
.114/4/14'
?
he above defined apparent. mnitude of the str if con1dered to be
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
0'71' ?
3,onsor
e 21,0 I. dem
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
eq.1.4.01). tgair4 s'ienCP 33C-. 41.e1difirthe rin ?
e ?Serint 4 s; Subdwarf
I
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
WAtfatvlAikfinfum14u.444evNisilkitioN4attia,,itmlA40,6.,v.:,-74,,,,,.w
4 I `
Declassified in Part -Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,.5 n't;Inq? Giy ' b 0 f r
r! 1 *) r i gi: I ? . 2? ,,I., :( : . 4: ti , 0 n V O
L1.114u
ilh 10 It he c fac:,.1...0'5e.i, that . It .. 1, vlz ,, ' 1.,:,:, th.o k a,,, It to Or .1f:'"5 I ,:?i ,1 3 ' the
v ,i,o...,,IA:".'
net,..Ite, ,)e , '0( ,..
0 TO
0 ad A, ',"`',. 6 r; . o', .,, 1'1, ?
PI ..1 ? ., 'r" 'tl.?' rrt., tl.ln,, a'e tz.nown
t., ( 3.
..,,; -,f ., i ...
., 's V.4^, r...i.,.o.,.:.1t; . ~ii,..t.,.7,.;),.on
Ill a. '4:era.r: 14'6.1..6 0 ,, .1 , . _ 1,,,,,,. ,.. ,J.....stvui'irrii,-,:.; w , Ipe Ilse .,, ,
,.. ,
't-'''s, 1:(1 4 l'''' of A.,i.),c, -,, 0 an lu * ,
..._
. _
- r.. 'ot .tlat,,,'...kat.1, V V, ,
. , ' , ' , ? , . :
- ? ,
,
..,
.?.
? , . .1.
l' ^...'. - '' ''' ' ' -,..
r
' t
*
The ener
&
I. 1-0
t Mn fcrrect hi ha
'
01,..4,1tiorl, ' tIrVil'''i kr:I?t1.1 ,SOrt I the eller
' rer7 ,,
- ' 0
VV V
i.) L., ' ' , j : ' f.. -.....? ''....'' '''' ,' ' , '
'? ?
?,ke
?
,
? ? ? ,
?
? ,,? ? - ?
r ? ? ? ? '
.? .? ? r
? , , " ? ?., '? ?
??,,, , ?? ? ,? ' ??
? 1 .
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
`;.
STAT
A
P"
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
r
i+,471
st/A.
.0,. -vf,
rg,
and ef tc0- v E.
.;
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
1,0
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,
fltre '1: , 1 a 1 zi4' c 1.,,.i 11 oe tiii4ket
...., , tb 0
ti, 1 0 ,
.; ? , 0 .
0,, ,t1: 4,11 f t
1:14,.at.,.7 -
1 ,?,. ''art,i t.; IL '
'r m1..0
1 le
,t ' 1 1 , 1 :
s thp 0 -
,.
"trio t. 0 Of eq:13 ''I'llb' ;jk 4: '.('' * 1
1; 1 t., .
'43 , ?
ti 1 ., (,. .1, 7
1 0 0 ,
f ,.'t t: -4
inaepe 1
41, v.. r
Ya
,,,4
' rei, 1.,
0
? if theeor enol."
SnC ',2-1
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
I ; ,
? ? '
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
cc)ntnt nterng.?,, 4./10 7 ant into the
and at (, , s .:Lus of the
-
ar ani con e
Lloor
eh
. 1
onstants a1, a2, ' ) he
los n
groups o
deteri.ne a unique ,
thea o , A. i
ofence osity upon is ti
0 and is g , e
upon L s upon anadd 0
Declassified in Pad-Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Cir, le o P 't , ' ' o .(,p,A. ti ,
r\ ' l'"tit4 '61 SO') i L CI
,
?, t.,,,,, i * irk dall, JIA 1 n i? ni
' ii,f?, . .1 c .6
r'''', .1'
ljt 41 ' r.. ',or...A. .'l A.1 , ...
' 41'( '...1,. la. th, ri. 1,, ,,,,,,
,,,,0iher
4
For the of Ung up the tk. r
p4 OC
t. 0. . 0 11 , ,
'8
aWS 9)
e a" 4
r ' or t so of Co aas'wnpt
tr ,
t,,11t1ivat10 of ons rcicLs the tr
corori. S
out, howevthat. pos b o ow. ,.? the single
,mgren, titan( b t thr Astroph/aik, Vol.7, 1936,p, 59; Erg. Ixaet ? iatur -
0 its
a. 9371, pa.ge 465, The expundizg of myi t the theory
found.ChdraekriitrxtucUon to the
MEMOS
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
446,0pitthaitgonragithligibliO t
or1atin(3,01
,ions the thy o
se of more
with the, h1p.:.:?07.the
nj
Qr in th seof th? pres atipr, or asaunpt ion
erti aunpti? sthen ',",c and g3, which are inolxxied in the
,
et _trttli,'r tnL14 IiflSCe citii-o4.44...4.0
0
t ooarld cond&tiofl8 (3.5) cn be rnodUied. If in the odfed od1tion
e do not dmen5i?uAls 1 conatant, then all the I3UC
4
ionsL ,t1 at 4t ?
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81701043R001200240001
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
lit";x).s.; et f7Q.Lit44?$ thAt 1.:17 or the Flos.e,,Li
,
" .f-i 474)
,
e dete ,, .1,, ,
n rtxL
d
'%
? ( es? -
5 a 11 that
4444 4
,ultrler we atl,--kute that
rc
? - Partv- 'et-.9r13. t P?f: t',14?? lnclePe' ndertt
?
:ha te.owt
s4.:,,,;ltertt t;,'.to rlecess:'4,t,,7 vast mane more
Ls
Jic to cOO only
44:1?4.s there arr-., uponr quant '4.'3 14, k. wal 44, 6voi ? .4.4 WILi.Ar.
6
'ere
" .
ionsif
.
1::.tp",,,lf,',)r,k 2 en tzl .?; 0.47,,,o11.11 tne de:f
4 4 ,
'ito4)1
-
STAT V
,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-R9P81-01043R001200240001-7
01131, 0 0 ,
?.
. ? . ..
.,. . .
.;,... ; . . . .
? . ... .? .,, ,,,, ,
';` It. .0,1;e, .'-trg'ri.t,t,1 1. ti..0 s ? .t.' t,',.,, 1,', 4t,id,, )-,'-i' w:' 1 ..arlf.1),.... rttlf3:1:1.es1.1 ,,,,6;:1?.174%)14.Nititle01.1,,$),.7,i , 1;:',ro r
''' ..: ' ' ..' '''''' ' '''' ' ' r ' . '' ' ' '.. . ' ' . " '' ' ' ' ' ' :' ' . ' ' : A '4'. r ' '''C. t:4:' ' :'11,41 411, i'V'T.. e, t..0 'r 3 a 1:,?!,,. 5,T7 '
., n '' vt.ItA?;60. of . .thfe ..;1...i.41i..e., i.7.1,f4.,. Lg.,..'.,')., . li.....(t) .: (r!,".? ,,,*-T, ....,),...,:.,,,, .,,,.. ? ,,. .... : . ... ?... . ,, . , .. . .. . . ,,
I:?
0 r .
'17.10,53 ? ,
rvi..1
e
. ,
t4o 4" 0
? .
vr, "Li r !cnfim. (r ? 44
j f2:, 0 4,1 'Al ,1?,1, ?S r?i(j VT:; k.447 ie
?
. .
.-.? ? -. ? . . ?
:41)".1c to 441):6 1equat? ?
con (3
?
,.?
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
_0.413%0444RIZ.,
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
A .1,1411,4i..t,tbo/iik,c 1.0 fr
414 1049 +,$1 ie.
,
Iztw ploviernirg e ekt13,0 ofcrU etl,er*(0,?,74t,
14 3.3.) -1,41 obtPoaitle(t 4.t rtittv'iv11,1AA' tvAse of rot.rfoliAlk
It Is not d141t ko pei4scisde ores(olf t,11,At t.)1.4't rorvAt1.11
ves itiot fol,11. Cr the Llase of tl)et 111,?Jfict1 ofa st,ztl,,,, tirt?t,f point source of enerki,?,y
tIrt, the sta,1.143 center, when the power of the point eoltre c)f enerb, is Lies'ir,ncild
a rbi t
FtP41%."0,21,11$ ( 3. ) arlet 0,1) ea, ? t out U ei tor tl.t1 octg ot 11.3,,,!,?,4), -
'vat . In 4:,,,''oTartaring theae forrAulas wlt.h $J41.drAriea or can 01',air,. ot.trta'
r
3,11g-s (3.2) alt,i A rc)r t.0,
? _
.
1.41t S 404.A
t
&Potion 1,,,ertain 61..tno:te ;?)o. ott lone ot rte
It. ,..et ... ,?", 7 , .1- om #, 4 ' .fr* ' '."' ,.. . %-, 1,.., 4 , .04,.... . 4, -
,
s,,,.e.1,,,, ..u., - .i..d.Q11.?44,?,....t.,or4s o 4 , 1.'4. 1.;,.1?.,..,..4. 1,,,,>, A 1441 i 0 4.
',..........7;,....n.......;,...,...................m................,--6., -? ,,,--- . ' , ....................
r . ?
r . '
In the tteo'r,t or the ()/Itbu;r3t of n" a* d'ur3n1.7 ''''''lle irlyeej.t4?r.'a't:iarl ?f t'h n?11-
,
_ .._s .. te raotiorts of gaseous vassi? or a sts,r it is tlec,essarl to 'utilize 'in t,he '-aitial
co /,,,,Lons data ort thecii,,,,,,,,r:1?0,114,;,..:Lori of tiletilaracter4.5-1;as; ,:-,,,t7 ,-;:rt,..!,,,,,,,..-
Iri the star drar,ing eccuilibriust. For thaq pto,rpose lttillzatlorl ol:' the solutions to
oyst,tra of ecluations (II) 1,41th tloundarY ecmdIttons (24'6) and (2*7) 1'5 irIcQnve-
, ,
ent bneem141:3iet::n tt,heo .t:Iticla.sP,lefx:rt: odreet:'':rs:e'la::::-f*tijcc:::.loar:of'eztclli:jrleo8-.the,, ''''''?(13,:t:',12'i,4?4:',r*:i'lect'',11,,,_ .
,
If t 1 d the s'attt SO v _ 4 addi ono. , 4
- t rs' it is kassf:1?a to (Ionsider ,the solutions to tile sYst,an 1 / ?
he rfe
lab,,e4 tlatiorls 04; tehcielii,astar. .. ,
i t to the bo:undarY ??nditi?1145 on '''' --brium (II) (in
o to the
oonsi er , . ii ej;r ort teri,a'.-TL
1
here we sha r
' ' 014itiC11115' ' ' 'F'. 4P + tir6 ?tar . - - _ Ped
,
on the
.
obtaining ari'' 6treCtiTe C'illti?n"1 (''''4'nCerrll'ne' non-Stead:I'. not) ore.
01 the Irarious
hIl treat the re le both13i_
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
0,
1
0 44hlch can
4,31?4
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
4 '41,0
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
,
41, titai1 r ';(.1 h h d i.enc upon.
Itt,t0
ettland ccQt11 ,
arrke,0 t ht, s
tive. Itt.orlot?o,-,te,,,? ?
:Sr.>
? 4
.41 Jo.i ' 7Lo
these ?cat13.wMre zrthtt
C 0:M
o
4.1,1 o L ,
at;
SI 1
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
s 7T,?????,????? ? ? ? ?
4 .. ir \17,;1',."64,.4k.,,,t ' "7,k V.,7, ,,t; . ? f k. ,. i'?? k-,?71,?7 ? ' ,17.11)- 7,.....,,,,, .;i 1,.. .!.. 1 - . '' ' .4i... ,., .4, I- ' :' ::. ? .' ,'',IY?. '7' ' ' r ? ' ... ? . ?. ,, ., ,
..;?,,,,,,.., ? q,,, 1-$..,- ? ,r.,1?, ?,4.. ?,, 'n',; ,,??? '1.-'7'.4, ., k;.'. 7 a. 7 ; '7,', .4 ' .,'I )47,,;7,1 ?? 7'. , ,r. ,,, ?'i ,,, . k l'''... ..c. -.ti '..') ? ? 4/ ,4 ' "(,;'i,1'., 0 fl7 ''''''''..',70!:,,t:t,,, ":',1i,:',!: 6 ' .t. ),).;,,,:,..,.. . . .
. ? . . , . , . . . . . ... , . . . . ., ? .. ,. , . . .
.. . . . ., . . , ? . ... . , , . .. ,
a t 1' 1 4 ..?r. '''. (.-.t. ?,.(`' ' A;,,14?'', .1 '11 ''', `'?.11'?'... 1 '', vH ' ' ? . , .. , . ? . . ..
, .
' '',.??.'", '' . '4., iv, '''. .4,;,.., . ? ';.' :,,.10..,-,4,? .!,....,. ,, , ', ,. , . ,,,,A. .. ',..., .,,,,,,, ,,,,,.., ', * , . , . . . , , ?
"t?JIU 01,1 U, 1 f
. , . . . . .
:4, 677k ,7.,??..;,k t ??77' k77 '1?
Y A
? , ? .? ? . ? ,
? ,? , , ?
?
, ? t 4
,
? it I str'*.(4e,
:1,1t1,2 4,00e?
...P. 4'. ?
? r'???,;.:;t(,::?-'
'
?
'
Declassified in Part - Sanitized Copy Approved for Release
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
?$t4' .!'" >. " 4
a'qb.vs.'t:;::k..t,i:kkt't.a
)1,3 &es e exp r
e arki thoxtt ofviwit op.1,0:te terR )5: wil?IrIttJQz.t..1.
'0110 az 0z ,A (II) h ir the, 01.
Declassified in Part
- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
A1t.
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
?
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
I 0,14,4145
\. 4.
es ts Pflarc f
'WO
another et on Where
correspond to the
i L8 by the t
epga?
(1,1 hat
0
th.ai the dis r ion
1 imanosit. This caae
from 'o
Declassified in
Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
.?
'
Urce it l.ows? t.i!t4t th t ratrIn.' t.,'-')Htaiacd.? as... a ..Con'?tai)t :relati've ...to the r-a.-
d 1us in th. s .zi:$44,1,..t.114',f0.1.10ing 'eqUatia6:: ? ''?-?
rir L 0, can be satifified ora,z,,,, for the a.s,,:rmption that
cad eqtation of the theor:,- of .radiatian
dz;
IT'1**tiffe
dr
can alidacys be satisfied and a:rip
fLulction of r, p and T*.
i'"0,??, the ,,o4 e,-ination of L ife ?
:known as
In part cuitr, If 0, then '? coast. In the 1,761-muelas (4?19) there
.to'ai 2
. . . .
. .
. . r
ee ill the iiel)ei.,41' "tIetvr'ee.n ?-'?,.'1.1e. Period.' al" Osci.1.1,a'tiol.-i of Br
, s 1,14mt . D
Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
t No: ? .0 A t.
44fit,p ? ?
? 4':4
, ??. .
?a w
'rrAhi eh
Ir. 0 . "1.11',31 e
?n S41
r413
4 . 5 1114
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part- Sanitized Copy Approved for Release 2012/,12/05 : ClA-RDp81-01043R001200240001-7
eo,ptiol.,t1; 413?no tAti.$ a 'ale, 0.!t1?41;r4rit ot trtot ort on th tqh.ol'o tilert on the tall
, ? , ,
? , , , . ? ,
,Ct t he StA '411 hCfN,1 t;;41,t1,t1 detiraing ,parzurtf,?11,4rs (5,2) nitt) da.r7ivt, thrz,t, ? tau .1?1
d:.i.tionr4.?tOnA..,41,,tiop.,..? 'then it rolicae ? -.from thi?
::' 1,,,,...1-?? ? 1.1????? 4 ?,,?? ''? ? : ' .
....."..4.it.vit ..,....: .,.. ....1,. V....r,..'4,,,...p...,..... ...
. . . . .. . . . . . 1,)04)
.. i'' r ? f
. . . ,
. . . .
. ?
??
.. . .. . . .
. . . .
. ....
.. . . . . . .. .
.. .? .. .. . ... . . . . . .. ..
......S11...ch???dt.o.?th4'a...3imnle? c..orcJidersrAtiOnucthat.. v?flmit On.,,,,,, wlt.h.6ut.....Oarl-v-in.?611t.'the?detail,ed
. .
? ? ?,.. :.??,.. .-. ?
.......... ........... . ..,... . .. . .. ? :. ? .? . . , . ? ? ? ?
... . .?
..rtztathfirat.i...,e4 p4..11..:?4.4;.....,:l. ? :,:.., o?? ...43,ubs.,,,.,.i.:.*1...f.:1?J,?0.t...i.,?!, for '...1,;11,;:ner . .17 ener,a1... .i,a,, a a 7,71.11::).t, .i.o.tvz . .t.1,,-?e.? r e 233.-L, i on ..
: (?.,.....,,,. 4 ) ,.;.,. : .,...,,,,,,,,?,-,7,...; . ..s,,,.,,,,,,-.3 . ..,,,, u. 74.?'?;1,4,;("?t..,..L.,4,, z3.r..101,1. ..,..,,:,?,......?,....r.?,:,?..,,,r,1,,.. zteans .?0,.f..'?eez.'!,,,I15?.z.c.;,724..,1..o..1).5e...r..-761t.:,:1,?0
. .,. .? . .
-
38
-tpi:.ica:1 Data ? the fiel,ation Pen. :,1..ty for
MiTererlt l'ffariables
1145?pori(-3-ki variabltBts A) regu1 var4J,Ables
cla) oelziheit:ta.
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
JA"/4 A ' 101 h 1, 401 C1 At,
;ALt4',4
l .1
07,716, .1) , ,
' '14;;,M1444 'j!?;& A.1!)Al.il
orgroups of , , id theYaiil aecor4Lrip; the, theo-
',I I rOrAlal, O. ?. ti i? 3 s, tion of the poixto , ,eh.,wers obta led. from the
ff' line
I, greement Id:01 the 1O Of the the()
et 'dotc s .. i le
hp,..t m h Yar tarat ons of
4) .,,n the
, 0 0 , r a 1, t
Rtlation k ts the
, $ ? .m. ii-lasson
'1,--rt..,, " " ..g-Pe ' '
Prr of the 1?01, ri t..;:lo 0 1,oa var...1, 4,'? i e V.',
v, i :4,,,,re arranped alor46, ce,'
?
3% es t the ,-01,-re, on
1-e for
r, lane t. 9 rld
t cerr
e tov
J.
in ay; ;
ffi of- flGV3O. a
ernovae eprese the.
Declassified in Part -_ Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
1".1) 14)14 04
?
k
tYr '11
to, 013 4 3t.rac atit,
rd t,e ermnt .31 4 :14
e ?rs of t
)ow 41P "r" active
in txzcorstats ectcring into
theA 4,
if the , on S., 'PM
?on oi the ga on the whole dcr1ned by th
yitem .1 ''''' he Lnti1 n41tior7 4,
I
Or.1, 4 by the CO
a, d
Impommom
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
ezi iii bru 03.
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
-
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
'43
dre f?A
!itsInto
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
WIIMINPInar
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05 : CIA-RDP81-01043R001200240001-7
:n.,441x14401A1Atifi'ffitakt.ft,
?pa
,ne
rd
7!:t
4,;;,.re. at.. rest. we. ?
n ent 4.insiot4rta
e
he Mock wave
a c f* iti ri a
Declassified in Part.- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
'
THAVINA,V,,,,M,1110.
?i(
0 or
?
chv.
wav
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
gft,
ocr
11 on
mut,
acc. r X
rT ed
01. ,;?p2.1"
vc4ritd ? rld
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
fhc 13 or th0 'Ll.I.a.t3 s(C2),
quant es
con ' ' 1, on .t'lle N ,, a o ac.' 1,1
...
th ront', tht 1.',1 k Pl' iVO tJe f.01 n. exp ,i,oro :
1' ?
fired b
t h the ord1tioj ) can be i.t1id a.Ca?chy data for
hegvten
of equatIo arter whichthe corstant Fdpha
Declassified in Part - Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
Declassified in Part- Sanitized Copy Approved for Release 2012/12/05: CIA-RDP81-01043R001200240001-7
our.1
ka. eus n In -
*1.4 ?
11.4tA 4 YehAt
.ra,'olial'old
4 -4, 47 vr,74' 111-A., A Vt. V, r'Aer,A A 4 5w; lx W.'s.. w
unction It, z satisfying ordinary diS. renUal oquatior4 last r
th the help of met ho