Bridgman PW-Dimensional Analysis

CO

>-

CO

164033

OSMANIA

UNIVERSITY LIBRARY

CallNo.

5

'

W

*L

Accession No,

3^0

9

Author Title .

DIMENSIONAL

ANALYSIS

BY

P.

W.

BRIDGMAN

PEOPE8SOE OF

PHYSICS IN

HARVARD

UNIVERSITY

NEW

HAVEN

YALE

UNIVERSITY PRESS

PrintedintheUnitedStates ofAmerica

First published,November19S2 Revisededition,March1931 Revisededition,second printing,October1937

Revisededition,third printing, April 1943 Revisededition,fourth printing,MarchJ946 Revisededition, fifthprinting,February1948 Revisededition,sixth printing,August1949

Allrights reserved. This bookmaynot be pro-duced, in whole or in part, in any form, ex-ceptby written permissionfromthepublishers.

PREFACE

THE

substance ofthe following pages

&&

rix&n as

a

series offive

lecturestothe

Graduate

Conference in_Physic^'of

Harvard

Univer-sity in thespringof 1920.

The

_growing

use ofthe

methods

of dimensionalanalysis in

tech-nical physics, as well as theimportanceofthe

method

in theoretical

investigations,

makes

it desirable that every physicist should

have

this

method

_{of analysis at his}

command.

There

is,however,

nowhere

a systematicexposition of the principles ofthe method.

Perhaps

the reason forthis lackis_{the feeling}that the subjectis so _simplethat

any

formal presentationis_superfluous.

There

do,nevertheless, exist

important misconceptions as to the

fundamental

character of the

method and

the detailsofitsuse.

These

misconceptions are so

wide-spread,

and have

so

profoundly

influenced the character of

_many

speculations, as I shall try to

show by

many

illustrative examples,

thatI

have

_thought

an

_attempt to

remove

themisconceptions well

worth

theeffort.

I_{have, therefore,}

_attempted

a systematicexposition of the

princi-plesunderlyingthe

method

ofdimensional_analysis,

and have

illus-trated the _applications with

many

examples

especially chosen to

emphasize the _points concerning

which

there is the

most

common

misunderstanding, such as the nature of a dimensional formula,

the

_proper

number

of

fundamental

units,

and

thenature of

dimen-sional constants.

In

addition to the

_examples

in the text, I

have

included at the

end

a

number

of practise problems,

which

I

_hope

willbe

found

instructive.

The

introductory chapteris addressedtothose

who

already

have

some

acquaintance withthegeneral method.

Probably most

readers

willbe of this class. I

have

tried to

show

in this chapter

by

actual

examples what

are the

most important

questionsin

need

of

discus-sion.

The

readerto

whom

_{the subject} is_entirely

new

_may

omit this

chapter without trouble.

I

am

under

_{especial obligation}to the_{papers of Dr.}

_Edgar

Buck-ingham

on

thissubject.I

am

also

much

indebtedtoMr.

M.

D.

_Hersey

ofthe

Bureau

of Standards,

who

a

number

of years ago presented Dr.

_Buckingham's

resultsto theConference in aseries oflectures.

PREFACE

TO

THE

REVISED

EDITION

IN reprintingthislittle bookafter eight years,only a few

modifica-tions,includingreferences to recentliterature,have been

found

nec-essary.

The

most

important isa

_change

in theproof of the

n

theo-rem

in

_Chapter

IV. I

am

much

indebted to Professor

Warren

Weaver

of the Universityof

Wisconsin

for calling

my

attention to

an

error in the original proof

and

also for _{supplying the}

form

of

the proof

now

given. I

have

also profited

from

several suggestions

of Dr.

_Edgar

_Buckingham,

although I

am

sure that Dr.

Bucking-ham

would

still_{take exception}to

_many

_things in the book.

An

appendix

has been

added

_containing the dimensional

formu-las of

many common

quantities as ordinarily defined. I

_hope

that

this will _prove useful inthe solution of actualproblems, but atthe

same

time I

_hope

that thepresentation of such a table will not

ob-scure the essential fact that the dimensionsthere given

have

about

them

nothingofthe absolute,butaremerelythose

which

experience has suggested as

most

likely to be of value in treating the larger

partof the

_problems

_{arising in practice.}

Since the first _printing of the

book

I

have

observedto

_my

_great

surprise that in spite of

what seemed

to

me

a lucid

and

convincing

exposition there arestill differences in

fundamental

points of view,

so_{that the subjectcannot yet}be_{regardedas entirely}

removed from

the realm _{of controversy.}

Nothing

thathas

_appeared

_{in these eight} years has caused

me

to

_modify

_my

original attitude,

which

is there-fore _{given again without change, but} inorder that the reader

_may

be able to

form

his

own

judgment,

a

few

of the

more

_important

ref-erences aregiven here inadditionto several in the

body

ofthetext.

NORMAN

_CAMPBELL,

_Physics,

The

Elements,

Cambridge

University

Press, 1920, Chapters

XIV

and

XV.

Phil.

_Mag.

1924, 1,1145, 1926.

Measurement

and

Calculation,

Longmans

Green

and

_{Co., 1928,}

Chapter

XIII.

J.

WALLOT, ZS.

f. Phys. 10, 329, 1922. E.

_BUCKINGHAM,

Phil.

_Mag.

48, 141, 1924.

MRS. T.

EHRENFEST-APANASSJEWA,

Phil.

Mag.

1,257, 1926.

(Refer-encetoMrs. Ehrenfest's importantearlier_paperswill be

found

in this article.)

P.

W.

_BRIDGMAN,

Phil.

_Mag.

2, 1263, 1926.

P.

W.

B.

Cambridge,

Massachusetts January, 1931

CONTENTS

PAGE

Preface

...

v

Prefaceto revised edition

...

vi

Chapter

I. Introductory

...

1

Chapter

II. Dimensional

Formulas

....

17

Chapter

III.

On

the

Use

of Dimensional

Formulas

in

Changing

Units 28

Chapter

IV.

The

n

Theorem

36

Chapter

V. Dimensional Constants

and

the

Number

of

Fundamental

Units

....

48

Chapter

VI.

Examples

Illustrative of

Dimensional

Analy-sis

...

56

Chapter

VII. Applications of Dimensional Analysis to

Model

Experiments.

Other

Engineering

Applications

...

81

Chapter

VIII. Applications to Theoretical Physics . . 88

Problems

107

Appendix

...

110

INTRODUCTORY

APPLICATIONS

of the

methods

of dimensional _analysis to _simple

problems, particularly inmechanics, are

made

by

every student of

physics.Let us analyze a

few

such

problems

inorderto refresh our

minds and

get before us

some

of the questions

which

must

be

answered

in acriticalexaminationof the processes

and

_assumptions

underlying the correct application of thegeneral method.

We

consider first the illustrative

_problem

used in _{nearly every}

introduction to this _subject, that of the simple

pendulum.

Our

endeavor

istofind,without going

through

adetailed solution of the

problem, certain relations

which must

be satisfied

_by

the various measurablequantities in

which

we

are interested.

The

usual

proce-dure

isas follows.

We

first

make

alistofall_{the quantities}on

which

the

answer

_may

be

_supposed

to

_depend;

we

then write

down

the

dimensions of_{these quantities,}

and

then

we

demand

that these

quan-titiesbe

combined

intoa functionalrelation insuch a

_way

that the

relation

remains

true

no

matter

what

thesize of the units in terms

of

which

the _{quantities are}measured.

Now

letustry

by

this

method

to find

how

thetimeof

swing

of the

simple

pendulum

depends on

the variables

which

determine the

behavior.

The

time of

swing

may

conceivably

depend on

the_length

ofthe

pendulum, on

its_mass,

on

_{the acceleration of gravity,}

and

on the amplitude ofswing.

Let

us write

down

the dimensions of these

variousquantities, usingforour

fundamental

systemof units mass,

length,

and

time. Inthe dimensionalformulasthe symbolsof mass,

length,

and

time willbe denoted

by

capitalletters, raised toproper

powers.

Our

list_{of quantities}isas follows:

Name

ofQuantity.

Symbol.

Dimensional

Formula.

Time

ofswing, t

T

Length

of

_pendulum,

1

L

Mass

of

pendulum,

m

M

Acceleration of gravity,

g

LT~

2

Angular

amplitudeofswing,

No

dimensions.

We

are to find t as a function of 1,

m,

g,

and

6, such that the

functional relation still holds

when

the size of the

fundamental

unitsis

_changed

in

_any

_way

whatever.

Suppose

that

we

have

found

this relation

and

write

t

=

f _(l,m,g,0).

Now

the dimensional formulas

show

how

the various

fundamental

units determine the numerical

_magnitude

of the variables.

The

numerical

_magnitude

of the time of

swing depends

only

on

thesize

of the unit of time,

and

isnot

_changed

when

the units of

mass

or

length are changed.

Hence

if the equation is to

remain

true

when

the units of

mass

and

_length are

changed

in

_any

_way

_whatever, the

quantities inside the functional sign

on

the right-hand side of the

equation

must

be

combined

insuch a

_way

_{that together}theyare also

unchanged

when

the units of

mass

and

length arechanged. In

par-ticular, they

must

be

_unchanged

when

the size of the unit of

mass

alone is _changed.

Now

the size of the unit of

mass

affects only the

magnitude

of the _quantity

m.

Hence

if

m

enters the

argument

of

the function at _all, the numerical value of the function will be

changed

when

the sizeof the

fundamental

unit of

mass

is_changed,

and

this

_change

cannot be

_compensated

_by

_any

corresponding

change

in the values of the other quantities, for these are not

affected

_by

_changesin thesizeof the unit of mass.

Hence

the

mass

cannot enter the functional relation at all. This

shows

that the

relationreducesto

t

=

f

_(U,*).

Now

1

and g must

_togetherenter thefunction insuch a

way

that the numerical

_magnitude

of the

argument

is

unchanged

when

the

size of the unit of length is

_{changed and}

the unit of time is _kept

constant.

That

is, the

change

in the numerical value of1

_produced

by

a

_change

in the size ofthe unit oflength

must

be exactly

com-pensated

by

the

_change

_produced

in

_{g by}

the

same

_change.

The

dimensional

formula

shows

that 1

must

be divided

_{by g}

for this to

be accomplished.

We

now

have

Now

a

_change

in the size of the

fundamental

units _produces

no

change

in the numerical

_magnitude

of the _{angular amplitude,}

be-causeitis_{dimensionless,}

and

hence 6

_may

enter the

unknown

such a

way

that the combination has the dimensions of T, since

these are the dimensions of t

which

stands alone

on

the left-hand

side _{of the equation.}

We

see

_by

_{inspection that 1/g}

must

enter as

thesquareroot inorder to

have

thedimensions _{of T,}

and

thefinal

resultistobe written

t=

V~Vg*

W

where

<t> is subject to

no

restriction as far as the present analysis

can go.

As

a matterof fact,

we know

from

elementary mechanics,

that<f> is

very

nearly a constant independentof 6,

and

is

approxi-mately equalto2*.

A

question

may

ariseinconnection with thedimensionsof6.

We

have

said thatit is_{dimensionless,}

and

thatitsnumerical

_magnitude

does not

_change

when

the size ofthe

fundamental

units of mass,

length, or time are changed. This ofcourse istrue, but it does not

follow that therefore the numerical

magnitude

of is uniquely

determined,as

we

see atonce

from

the possibility of

measuring

in

degrees or in radians.

Are

we

therefore justified in treating 6 as a

constant

and

sayingthatit

_may

enter the functional relation in

any

way

whatever?

Now

let us discuss

_by

the

same method

of analysis the time of

small oscillation of a small

_drop

of liquid

under

its

own

surface

tension.

The

_drop

isto_{be thoughtof as entirely outside the}

gravita-tionalfield,

and

theoscillations refer toperiodic changes offigure,

as

from

_spherical to _ellipsoidal

and

back.

The

time of oscillation

will _evidently

_{depend on}

the surface tension of the liquid,

on

the

density of the liquid,

and on

theradius of theundisturbed sphere.

We

have, as before,

Name

ofQuantity.

Symbol.

Dimensional

Formula.

Time

of oscillation, t

T

Surface _tension, s

MT~

2

Density of_liquid,

d

MLr

3

Radius

of drop, r

L

We

are tofind f such that

t

=

f (s,d,r)

where

f issuch that this relation holds true numerically

whatever

measured.

The

method

is _{exactly the}

same

as for the

_pendulum

problem. Itisobvious that

M

must

cancel

from

the right-handside

ofthe equation. This

can

occur only ifs

and d

enter

_through

their

quotient.

Hence

t

=

_f(s/d,r).

Now

since

L

does not enter t,

L

cannot enter f.

Hence

s/d

and

r

must

be

combined

insucha

_way

that

L

cancels. Since

L

enterss/d

tothethirdpower,it isobviousthats/d

must

be divided

_by

thecube

of r in orderto_{get rid of L.}

Hence

t f _(s/drs).

Now

the dimensions of s/dr8 are

T~

2

.

The

function

must

be of

such a

form

that these dimensionsare converted into T,

which

are

thedimensionsof the left-handside.

Hence

thefinalresultis t

=

Const

_V

dr8

/s.

That

is, the time ofoscillation is proportional to the three halves

power

of the radius,tothesquare root of the density,

and

_inversely

tothe squareroot of the surface tension. Thisresultis checked

by

experiment.

The

result

was

given

by Lord

Rayleigh as

_problem

7 inhis

_paper

inNature,95, 66, 1915.

Now

let us_{stop to} ask

what

we

meant

when

in the beginning

we

said that the timeofoscillation will

_''depend"

only

on

thesurface

tension, density,

and

radius.

Did

we mean

that the results are

inde-pendent

of the atomic structure of the liquid, for

example

?

Every-onewill

admit

thatsurface tensionis

due

tothe forces

between

the

atoms

in the surface _{layer of the liquid,}

and

will

depend

in a

_way

too complicated for us at present to _{exactly express}

on

the shape

and

_{constitution of the atoms,}

and

on

the nature of the forces

be-tween them. If thisis true,

why

should notall theelements

which

determine the forces betweenthe

atoms

also enterouranalysis, for

they are certainly effective in _determining the physical behavior?

We

might

justify our procedure

by some

such

answer

asthis.

"Al-though

it is true that the behavior is determined

by

a

most

com-plicatedsystem ofatomic _forces, itwill be

found

that these forces

affect the result only in so far as they conspire to determine one property, the surface tension." This _{implies that} if

we

were

to

as

much

as

we

pleasedin atomic properties,

we

would

find thatall

drops

of the

same

radius, density,

and

surface tension, executed

their oscillations in the

same

time.

We

add

that the truth of this

reply

may

be checked

_{by an}

appeal to experiment.

But

our critic

may

noteven yet besatisfied.

He may

ask

how

we

were

sure

before-hand

that

among

thevarious_{properties of the liquids of}

which

the

drop might

be

composed

thesurface tension

was

the onlyproperty

affecting the time of oscillation. It

may

seem

quite conceivable to

him

that the time of oscillation

might depend on

the viscosity or

compressibility,

and

if

we

are compelled to appeal to experiment,

of

what

value is

our

dimensional analysis?

To

which

we

would

be

forced to _reply that

we

have

indeed

had

a

wider

experimental experience

than

our critic,

and

that there are conditions

under

which

the time of oscillation does

_{depend on}

the viscosityor

com-pressibility in addition to the surface tension, but that it will be

found

as a matterof

_experiment

that ifthe radius of the

_drop

is

made

smaller

and

smaller there is a point

beyond which

the

com-pressibilitywill be

found

to_play

an

imperceptibly small part,

and

in the

same

_way

if _{the viscosity} ofthe liquid is

made

smaller

and

smaller, there willalso _{be a point}

beyond which any

further

reduc-tion of the viscosity will _{not perceptibly} affectthe oscillation time.

And

we

add

thatit istosuch conditions as these that our _analysis

applies. Instead of appealing to direct _{experiment to justify}

our

assertions,

we

might, since

our

critic is

an

intelligent critic, appeal

to that generalization

from

much

_{experiment contained}inthe

equa-tions of _{hydrodynamics,}

and

_{show by}

a _{detailed application of the}

equations tothe _present

_problem

_{that oompressibility}

and

viscosity

may

be neglected

beyond

certain limiting conditions.

We

shall _{thus ultimately be} able tosatisfy ourcritic ofthe

cor-rectness ofour procedure, but todo it _requiresaconsiderable

back-ground

of physical experience,

and

the exercise of a discreet judg-ment.

The

_{untutored savage} in the bushes

would

probably not be able to

_apply

the

methods

of dimensional_{analysis to} this

_problem

and

obtain results

which would

satisfy us.

Now

let us consider

a

third

problem

Given two

bodiesof masses

m

t

and

m

2in

empty

space, revolving about each other ina circular

orbit

under

their

mutual

_{gravitational attraction.}

We

wish to find

how

thetimeof revolution

_depends

ontheothervariables.

We

make

Name

ofQuantity.

Symbol.

Dimensional Formula.

Mass

offirst _body,

_n^

M

Mass

of_{second body,}

m

2

M

Distance _{of separation,} r

L

Time

of revolution, t

T

These are evidentlyallthe quantities physically involved,because

whenever

we

compel two

bodies of masses

m

l

and

m

2to describe

a

circular orbitabout each other

under

their

own

gravitational

attrac-tion in

_empty

_space at a distance ofseparation r,

we

find that the

time ofrevolution is _always thesame,

no

matter

what

thematerial of

which

the bodies are composed, or theirpast history, chemical,

dynamical, or otherwise.

Now

letus search for the functional rela-tion,writing,

t

=

f

_(m^

m

2,r).

We

demand

thatthisshallholdirrespectiveofthesize ofthe funda-mental units.

A

moment's

examination confuses us, because the

left-hand side involves only the element_{of time,}

and

the elements

of the right-handside do not involve thetime at all.

Our

critic at

our elbow

now

suggests, * *

But

you have

not includedalltheelements

on which

the result _depends; it is obvious that

_{you have}

left out

the _{gravitational} constant.

"

_"But,"

say we,

"how

can this be?

The

_{gravitational} constant can look out for itself.

Nature

attends

tothat forus. Itisundeniablethat

two

bodies ofthemasses

m

_l

and

m

2

when

placed at a distance r apart

always

revolve in the

same

time.

We

have

included all the _{physical quantities}

which

can be

varied.

"

But

our

critic insists,

and

to oblige

him

we

try the effect

of_{includingthe gravitational} constant

_among

the variables.

We

call

the _{gravitational} constant

G;

it _obviously has the dimensions

M"

1

_L

8

_T~

2

, since it isdefined

by

the equation ofthe force

between

nil

^2

two

gravitating bodies,

forces

G

.

A

"constant"

which

has

r2

dimensions

and

therefore changes in numerical

_magnitude

when

the

fze

of the

fundamental

unitschangesiscalled

a

"dimensional"

constant.

We

now

have

to find a function such that the following

relationissatisfied:

t

=

f

_(m^m^r).

Now

thisfunctional equationisnot_{quite soeasy} tosolve

by

inspec-tion as the

two

previous ones,

and

we

shall have to use a little

algebra

on

it. Let us suppose thatthe function is _expressedin the

form

ofa

sum

of products of

_powers

of the arguments.

Then we

know

that if the

two

sides of the equation are to

remain

equal no

matter

how

the

fundamental

units are

changed

in _magnitude, the

dimensionsof _{every one}of the_{product terms}

on

the right-handside

must

be the

same

as those of the left-handside, that is, the

dimen-sions

must

be T.

Assume

thatatypical product

term

isof the

form

m*

m*

G>

r6

.

This

must

havethedimensionsof T.

That

is,

M

M*

(M-

1

L'T-')>

L

=

T.

Writing

down

the conditions

on

theexponents gives

Hence

The

values of a

and

/? are not uniquely determined, but only a

relationbetween

them

isfixed. This isas

we

shouldexpect, because

we

had

_onlythree equationsof condition,

and

four

unknown

quan-titiesto_satisfy

them

with.

The

relationbetweena

and

ft

shows

that

m

t

and

m

2

must

enter in the

form

m~

4 ( J ) ,

where

1 there is

no

-4/m.y

2 1 I I

\J

restriction

on

the valueofx.

Hence

our

unknown

functionisofthe

form

f

=

2

A,

-J-where x

and A,

may

have

any

arbitrary values.

We

may

rewrite f,

by

factoring, in the

form

But

now

₂

A

x ( -} , if there is

no

restriction

on x

or A,, is

DIMENSIONAL ANALYSIS

merely an

arbitrary function of m^/m,,

which

we

writeas <f>(

^?

Hence

our

finalresultis t

=

G*m}

That

is, the square of the periodic time is proportional to the

cube _{of the distance of separation,}

and

_{inversely as the}

gravita-tional constant, other things beingequal.

We

can,

by

other_argument,find

what

thenatureof the

unknown

function

_^

isinone_special case.

_Suppose

a very

heavy

centralbody,

and

asatellite so light that the

two

togetherrevolve

_{approximately}

aboutthe centerof the

heavy

body. It is obvious that

under

these

conditions the timeof revolution is_independentof the

mass

of the

satellite, for if its

mass

is doubled, the attractive force is also

doubled,

and

double the force_acting

on

double the

mass

leaves the

acceleration,

and

so the time of revolution, unaltered. Therefore

under

these special conditions the

unknown

function reduces to a

constant, if

we

denote

_by

m

x the

mass

of the satellite,

and

the

rela-tion

becomes

t

=

Const r .

G*

nil

Thisrelation

we know

is verified

_by

the factsofastronomy.

Our

critic _{seems, therefore, to}

have

been_justified

_by

the results,

and

we

should have included the _{gravitational} constant in the

original list.

We

are neverthelessleftwith

an

uncomfortablefeeling

because

we

do notsee quite

what was

the matter with our

argument,

and

we

are disturbed

_by

the foreboding that at

some

time in the

future there

_may

perhaps

be a dimensional constant

which

we

are

not clever

_enough

to think of,

and

which

may

not proclaim the

impossibility of neglecting it in _quite such

uncompromising

tones

as _{the gravitational}constantinthe example.

We

are afraid that in

such acase

we

will_{get the incorrect} answer,

and

not

know

it until

a

Quebec

bridgefalls

down.

Besidethematter of dimensional constantsthelast

_problem

brings

up

another question.

Why

is it that

we

had

to

assume

that the

un-known

function could be representedasa

sum

ofproductso

%

f

_powers

of the independent variables? Certainly there are functions in

be arbitrarily restricted toa small partof the functions

which

one

ofher

own

creaturesisable toconceive?

Consider

now

a

_{fourth problem,} treated

by Lord

Rayleigh in

Nature,vol. _{15, 66,} 1915. This is a rather

_{famous problem}

in heat

transfer, treated before Rayleigh

by

Boussinesq.

A

solid body, of

definite geometrical shape, but variable absolute dimensions, is

fixed in a stream of liquid,

and

maintained at

a

definite

tempera-turehigher thanthetemperature ofthe liquid at pointsremote

from

thebody. Itis_requiredtofindthe rate at

which

heat istransferred

from

the

body

totheliquid.

As

before,

we make

alistof thevarious

quantities involved,

and

their dimensions.

Name

ofQuantity.

Symbol.

Dimensional

Formula.

Rate

ofheattransfer,

h

HT-

1

Linear dimension of body,

a

L

Velocity ofstream,

v

LT~

l

Temperature

difference, 6

Heat

_{capacity of liquid per}

unit volume, c

HL~

80-1

Thermal

_{conductivity ofliquid,}

k

HL~

T"

1

0~*

This isthe first heat

_{problem which}

we

have met,

and

we

have

introduced

two

new

fundamental

units, a quantity of heat

(H),

and

a unit of_temperature (0). It is tobe noticed that the unit of

mass

does not enter into the dimensional formulas of

any

of the

quantities inthisproblem. If

we

desired,

we

might

have introduced

it, dispensing with

H

in so doing.

Now,

just as in the lastexample,

we

suppose that the rate of heat transfer,

which

isthe _quantity in

which

we

are interested,is_expressedasa

sum

ofproductsof

powers

ofthe arguments,

and

we

writeoneof the typicalterms

Const

a"0*v*c* k'.

As

before,

we

write

down

theconditions

on

theexponents

imposed

by

the_requirementthatthedimensions ofthis_productarethe

same

as those of h.

We

shall thus obtain four _equations, because there

arefour

fundamental

kindsof unit.

The

equationsare

8

+

e

=

1 condition

on exponent

of

H

p

_

_

o condition

on exponent

of o

+

y 38 e condition

_{on exponent}

of

L

y e

=

1 condition

_{on exponent}

of

T

ofthe

unknowns

must remain

arbitrary.

Choose

thisonetobey,

and

solvetheequationsintermsofy. Thisgives

/?=!

Hence

the

_{product term}

above

becomes

Const

a

0k

( )

V

k

/

The

_complete solution is the

sum

ofterms of this type.

As

before,

thereis

no

restriction

on

theconstant or

on

y,sothatalltheseterms

togethercoalesce into asingle arbitrary function, giving the result

^

/acv\

h

=

_ka0F[~-~

.

\ fc _/

Hence

the rateof heattransferisproportionaltothetemperature

difference, but

depends on

the other _{quantities in} a

_way

not

com-pletely specifiable.

Although

the

form

of the function

F

is not

known

neverthelessthe

form

of the

_argument

of the function

con-tains _{very valuable information.}

For

_instance,

we

are

informed

that the effect of

_changing

the velocity of the fluid is _{exactly the}

same

asthat of

changing

itsheat_{capacity. If}

we

double_{the velocity,}

keepingtheothervariablesfixed,

we

affect the rate ofheattransfer

precisely as

we

would

if

we

doubled the_{heat capacity of}theliquid,

keepingtheothervariables constant.

This problem, again, is _{capable of raising}

_many

_questions.

One

of these questionshas been raised

_by

D.

_{Riabouchinsky}

in _Nature,

95,591, 1915.

We

quoteas follows.

In

Nature

of

March

18 I^ord _{Rayleigh gives} this formula,

h

=

k a

F

/

V

_{consideringheat,temperature,}

length,

and

time

as four

"

_independent

"

quantities.

If

we

_{suppose only} three of these quantities are "really

inde-pendent,"

we

obtainadifferentresult.

For

example, ifthe

tempera-ture is defined as the

mean

kinetic _energy of the molecules, the

principle of similitude* allowsus onlyto affirmthat

Vk

*

Rayleighand other English authors usethisnamefordimensional analysis.

-,ca

V

a2'

That

is, instead ofobtaining a result with

an

unknown

function

of _{only one argument,}

we

should have obtained a function of

two

arguments.

Now

afunctionof

_{two arguments}

isof_{course very}

much

lessrestricted initscharacterthanafunctionofonly one argument.

For

instance, ifthe function isof the

form

_suggested in

two

argu-ments, it

would

not follow at all that the effect of

_changing

the

velocity is the

same

as that of

_changing

the heat _capacity.

Ria-bouchinsky,therefore,

makes

areal point.

Lord

_Rayleigh replies to

Riabouchinsky

as follows

_{on page}

644

of the

same volume

ofNature.

The

_question raised

by

Dr.

_{Riabouchinsky}

belongs rather tothe

logic than theuse of the principle of similitude, with

which

I

was

mainly

concerned. It

would

be well

_worthy

of further discussion.

The

conclusion thatI_gave followsonthe basis of theusualFourier equationsfor theconductionof heat, in

which

temperature

and

heat

are _regarded as _{sui generis.} It

would

indeed be a

_paradox

if the

further

_knowledge

of the nature of heat afforded _{us by molecular}

theory put us in

a

vorse _position than before in _{dealing with a}

particularproblem.

The

solution

would seem

to be that the Fourier

equations

embody

something as to the nature ofheat

and

tempera-ture

which

is _ignored in the alternative

argument

of Dr.

Ria-bouchinsky.

This reply of

Lord

Rayleigh is, I think, likely to leave us cold.

Of

course

we

do

not questiontheabilityof

Lord

Rayleigh to obtain

the correct result

by

the use of dimensional analysis, but

must

we

have the experience

and

physical intuition of

Lord

Rayleigh to

obtain the correct result also?

Might

not perhaps a little

examina-tion ofthelogicof the

method

of dimensional analysisenable usto

tell

whether

_temperature

and

heat are ''

really

''

independent units

or not,

and what

the

proper

way

of _{choosing our}

fundamental

unitsisT

Besidethe

prime

questionofthe_proper

number

ofunitstochoose

in _{writing our dimensional formulas,} this

_problem

ofheat transfer

raises

_many

others also of a

more

_{physical nature.}

For

instance,

why

are

we

justified in _neglectingthe density, or the viscosity, or

the compressibility, or the thermal expansion of the liquid, or the

absolute temperature?

We

will _probably find ourselves able to

justify the neglect of all_{these quantities,} but the justification will

involve real

_{argument and}

a considerable physical experience with

problem

cannot be solved

by

the philosopher in his armchair, but

the

knowledge

involved

was

_{gathered only}

_{by someone}

at

some

time

soiling his

hands

withdirect contact.

Finally,

we

considera fifthproblem, of

somewhat

different

char-acter. Let usfind

how

theelectromagnetic

mass

of_{a charge}of elec-tricity uniformly distributed throughout a sphere

depends on

the

radius of the sphere

and

the

amount

of the charge.

The

_charge is

considered tobe in

_empty

space, so that the

amount

of the charge

and

the radius of the sphere are the only variables.

We

_apply

the

method

already used.

The

dimensions of the charge (expressed in

electrostatic units)

we

get

from

thedefinition,

which

statesthat the

numbers measuring

the

_magnitude

of

two

chargesshallbesuchthat

the force between

them

is equal to their _{product divided}

_by

the

square of the distance

between

them.

We

_accordingly have the

following table.

Name

ofQuantity.

Symbol.

Dimensional

Formula.

Charge, e

_M*

_L*

_T

!

Radius

of sphere, r

L

Electromagnetic mass,

m

M

We

now

write

m

f (e,r)

and

trytofind the

form

offso thatthis relationis_independent of

the sizeof the

fundamental

units. It isobviousthat

T

cannotenter

the right

hand

sideof the equation, since it does not enter the left,

and

since

T

enters the right-hand side _{only through e,} e cannot

enter.

But

ifedoes not enter theright-handside of the equation,

M

cannot enter either because

M

enters only into e.

Hence

we

are

left with a contradiction in requirements

which shows

that the

problem

is_{impossible of solution.}

But

_{here again our} Mephistophe-lean critic suggests that

we

have left out a dimensional constant.

We

demur

; our systemisin

empty

space,

and

how

can

empty

space

require dimensional constants?

But

our

critic insists that

_empty

space does have properties,

and

when we

push

him, suggests that

light is _{propagated with a} definite

and

characteristic velocity.

So

ue

try again, including the velocity oflight,c, ofdimensions

LT"

1

,

and

now we

have

We

now

no

longer encounter the previous difficulty, but

imme-diately, with the help of

our

experience with

more

complicated

examples,findthe solution tobe

ea

m

=

Const

.

re*

This

formula

_may

beverified

_{from any book on}

_{electrodynamics,}

and

our

criticis again justified.

We

worry

overthe matter of the dimensional constant,

and

ultimately take

some

comfort

on

recol-lecting that cisalsotheratio of the electrostatic to the

electromag-netic_units, butstill it isnot veryclear to us

_why

this ratio should

enter.

On

reflecting

on

the solutions of the

problems

above,

we

are

troubled

by

yet anotherquestion.

Why

is itthat

an

equation

which

correctly describesa relation

between

various measurable physical

quantities

must

inits

form

be

_independent

ofthesizeof the

funda-mental

units?

There

does not

seem

to be

_any

_{necessity for} this in

thenature ofthe

measuring

processitself.

An

equationis

a

descrip-tion of

a phenomenon,

or classof

_phenomena.

It is

a

statement in

compact form

that if

we

_{operate with a physical}

_phenomenon

in

certainprescribed

ways

soas to obtaina setof

numbers

describing

the results of the operations, these

numbers

will satisfy the

equa-tion

when

substituted into it.

For

instance, let _{us suppose}

our-selves in_{the position}ofGalileo,tryingtodeterminethe

law

of

fall-ing bodies.

The

material of

our

observation isall _{the freely falling}

bodies available at the surface ofthe earth.

We

use as our

instru-ments

of

measurement

a certainunitof length,letus saytheyard,

and

acertainunit oftime,let_{us say}theminute.

With

these

instru-ments

we

operate

on

all _{falling bodies according} to definite rules.

That

is,

we

obtain all the pairs of

numbers

we

can by

associating

for

_{any and}

all of the bodies the distance

which

it has fallen

from

restwith the intervalof time

which

has elapsedsince it started to

fall.

And

we make

_{a great discovery} in the observation that the

number

expressingthe distance offallof

any

body,

no

matter

what

itssizeor_{physical properties}orthe distanceithasfallen, is_always

a fixed constant factor timesthe square of the

number

expressing

the corresponding elapsed time.

The numbers which

we

have

ob-tained

by measurement

to fit into this _relationship

were

obtained

description,

and

our discoveryis

an

important discovery even

under

the restriction that distance

and

time are to be

measured

withthe

same

_{particular units as those}

which

we

originally employed.

We

can write

our

discovery in the

form

of

an

equation

s=.Constt2.

Now

an

inhabitant of

some

other _country,

who

uses

some

other system ofunits equally as unscientific asthe

yard and

the minute, hearsof our discovery,

and

tries

our

experiments withhis measur-ing instruments.

He

verifies our result, except that he

must

use a

different factor of proportionality in the equation.

That

is, the

constant

depends on

thesize of the unitsused inthe _{measurements,}

or in other words,is a dimensionalconstant.

The

verification of _{our discovery}

_{by an}

inhabitant of another

country is _reported to _us,

and

we

retire to _contemplate.

We

at

length offer the

comment

that this is as it should be,

and

that it

could not well be otherwise.

We

offer _{to predict in}

advance

just

how

the constant shouldbe

_changed

tofitwith

_any

_systemof meas-urement,

and on

being asked for details,

make

the sophisticated

suggestion that

we

so

_change

the constant _{as to exactly neutralize}

any change

inthe

numbers

representing thelength or the time, so that

we

will still have essentially the

same

equation as before. In

particular, ifthe unitof lengthis

made

half as large as originally,

so that the

number

measuring

a certain distance of fall is

now

twice as large asit

was

_formerly,

we

_multiplytheconstant

by

2 so

as to

_compensate

for the factor 2

_{by which}

otherwise the left-hand

side of theequation

would

betoo_{large.Similarly}iftheunitoftime

is

made

threetimesaslongas formerly, so that the

number

express-ing the duration of a certain free fall

becomes

only

%

of its

original value, then

we

will multiply the constant

_by

9 to

com-pensate for the factor

_{%, by which}

otherwise theright-hand sideof the_equation

would

be too small. In other words,

we

givetothe

con-stant the dimensionsof_{plus one} in length,

and minus two

in time,

and

so obtain a

formula

valid

no

matter

what

the size of the

fundamental

units.

This experience

emboldens

_us,

and

we

try its success with other

much

more

complicatedsystems.

For

instance,

we make

observations of the height of the tides at our nearest port, using a foot rule to

thetime-measuring instrument.

As

the result of

many

observations,

we

findthattheheightofwater

_may

_{be represented}

_by

the

formula

h

=

5 sin0.50661.

We

now

write this in a

form

to

which any

other observer using

any

other_system of units

_may

alsofit his

measurements by

intro-ducing

two

dimensional constants into the formula,

which

takes

the

form

h

_G!

=

5sin0.5066

C

21,

where

C^ has the dimensions of

Lr

1

,

and

C

2 has the dimensions

of

T-

1

.

This result _{immediately suggests} a _{generalization.}

_Any

_equation

whatever,

no

matter

what

its _form,

which

correctly reproduces the

results of

measurements

made

with

any

particular system of units

on any

physical system,

may

be

thrown

intosucha

form

thatit will

bevalid for

measurements

made

withunits of differentsizes,

by

the

simpledevice ofintroducingasafactorwith each observed quantity a dimensional constantofdimensions _{the reciprocal of those of the}

factor beside

which

it stands,

and

ofsuch a numerical value that in

the originalsystemof unitsithasthe value unity.

Of

course it

_may

often

happen

that the

form

of the equation is

suchthat

two

or

more

of thesedimensionalconstants coalesce into a

single factor.

The

first

_example

above ofthe falling

body

is one of

this kind.

The

_{general rule just given}

would

have led to the

intro-ductionof

two

dimensional constants, one with s

on

the left-hand

side of _{the equation,}

and

the other with t2

on

the right-hand side

ofthe equation,but

_by

multiplyingup, these

two

may

be

combined

intoasingle constant.

Our

query

istherefore answered,

and

we

see that every equation

can

be

_put

in such a

form

thatitholds

no

matter

what

the size of the

fundamental

units, but

we

areleftin a _greater

_quandary

than ever _{with regard} to dimensional constants.

_May

there not be

new

dimensional constants appropriate to every

new

kind of _problem,

and

how

can

we

tellbeforehand

what

thedimensionalconstantswill

bet If

we

cannottell beforehand

what

dimensional constants enter

a problem,

how

can

we

hope

to

_apply

dimensional _analysis?

The

dimensionalsituation_{thus appears even}

more

_hopelessthanat_first, for

we

could see a sort of reason

_why

the gravitational constant should enter the

_problem

of

two

_revolving bodies,

and

could even

catch

a glimmer

of reasonableness in the entrance of the velocity

of light into the

problem

of electromagnetic mass,

but

it is

cer-tainly difficult to discover reasonableness or _{predictableness} if

dimensional constants

can

be

used

indiscriminatelyas factors

by

the sideofevery

measured

quantity.

In

our considerationof the

problems

above

we

havealso

made

one

more

observationthatcallsfor

comment.

We

have

noticed that every dimensional

formula

of every measurable quantity has always involvedthe

fundamental

units asproducts of powers. Isthis

neces-sary,or

may

therebeotherkindsofdimensional formulasfor

quan-tities

measured

in_{other ways,}

and

ifso,

how

will

our

methods apply

to suchquantities?

To

sum

up,

we

have

met

inthisintroductory chapter a

number

of

_important

_questions

which

we

must answer

before

we

can

hope

to use the

methods

ofdimensional_analysis with

_any

_{certainty that}

ourresultsarecorrect. Thesequestionsareas follows.

First

and

foremost,

when

do

dimensional constants enter,

and

what

istheir

form

?

Is it _necessary that the dimensional

formula

of every

measured

quantity bethe

_product

of

_powers

ofthe

fundamental

kindsofunit?

What

isthe

meaning

of quantitieswith

no

dimensions?

Must

thefunctions descriptive of

phenomena

be restricted tothe sura ofproducts of

powers

of the variables?

What

kinds of quantity should

we

choose asthe

fundamentals

in

terms of

which

to

measure

the others? In _particular,

how

_many

kindsof

fundamental

units are there?Isit_{legitimate to}reduce the

number

of

fundamental

units as far as possible

by

the introduction

ofdefinitions in _{accord with experimental} facts?

Finally,

what

is the criterion for _{neglecting a} certain kind of

quantity in

_any

_problem, as for

_example

_{the viscosity} in the heat

flow _problem,

and

what

is thecharacterof the result

which

we

will get,'approximate or exact?

And

if approximate,

how

good

is the

DIMENSIONAL

FORMULAS

IN

the introductory chapter

we

considered

some

special

problems

which

raised a

number

ofquestions that

must

be

answered

before

we

can

hope

to really master the

method

of dimensional analysis.

Let

us

now

begin the formal

_development

of the subject, keeping

these questions in

mind

tobe

answered

as

we

proceed.

The

_purpose of dimensional _analysis is to give certain informa-tion aboutthe relations

which

hold between the measurable

quanti-ties associated with various

_phenomena.

The

advantage

of the

method

is that it is rapid; it enables us to dispense with

making

a complete analysis of the situation such as

would

be involved in

writing

down

the equations of motion of a mechanical system, for example, but

on

the other

hand

it does not _{give as complete}

infor-mation

as

might

beobtained

_by

carryingthrough adetailed analysis.

Let us in the first place consider thenature ofthe relations

be-tween

themeasurablequantities in

which

we

are interested. In

deal-ing with

any

phenomenon

or

_group

of

_phenomena

our

method

is

somewhat

as follows.

We

first

measure

_{certain quantities}

which

we

have some

reason to _{expect are} of _importance in _describing the

phenomenon.

These quantities

which

we

measure

are of different

kinds,

and

for each different kind of quantity

we

have

a different

rule of operation

by which

we

measure

it, that is, associate the

quantity with a

number.

_Having

obtained

a

sufficient array of

numbers by which

the different quantities are measured,

we

search

for relations

between

these

numbers, and

if

we

are skillful

and

fortunate,

we

find relations

which

can-be expressedin mathematical

form.

We

are usually interested preeminently in one of the

measured

quantities

and

try tofinditinterms of the others.

Under

such conditions

we

would

searchforarelationofthe

form

x

x

~f

(x2,x?,x4,etc.)

where

x,, x2, etc., stand for the

numbers which

are the measures

of particular kinds _{of physical quantity.}

Thus

x

the

number

which

isthe

measure

ofavelocity,

x

2

may

standforthe

number

which

isthe

measure

ofaviscosity, etc.

_By

a sortof

short-hand method

of statement

we

_may

abbreviate this long-winded

description intosaying thatXL isa velocity, butofcourse it really

is_not,butis_{only a}

number

which measures

velocity.

Now

the first observation

which

we make

with regard toa

func-tional relation like the above is that the

arguments

fall into

two

groups,

depending on

the

_way

in

which

the

numbers

are obtained

physically.

The

first

_group

of quantities

we

call

primary

quantities.

These are the quantities which, according to _{the particular} set of

rules of _operation

_{by which}

we

assign

numbers

characteristic of

the

_phenomenon,

areregardedas

fundamental and

of

an

irreducible

simplicity.

Thus

in the ordinary systemsof mechanics, the

funda-mental quantities are taken as mass, length,

and

time. In

_any

functional relation such as the above, certain

arguments

of the

function

_may

be the

numbers which

are the

measure

of certain

lengths, masses, or times.

Such

quantities

we

will agree to call

primary

quantities.

Inthe

measurement

of

primary

quantities, certain rulesof

opera-tion

must

be setup, establishing the physicalprocedure

by which

it

ispossibleto

measure

a lengthintermsofa_{particularlength}

which

we

choose as the unit oflength, or

a

time in terms ofa particular

intervaloftimeselected asthe standard, orin general, itis

charac-teristic of

primary

quantities that there are certain rules of

proce-dure

by which

it is _possible to

measure

any primary

quantity

directly interms ofunitsofits

own

kind.

Now

itwill be

found

that

we

always

make

atacit requirement inselecting the rules of

opera-tion

_{by which primary}

_{quantities are}

measured

in terms of

quanti-tiesoftheir

own

kind.This requirementfor

measurement

of length,

for_example,isthatifa

new

unit of_length ischosenlet_{us say}half the length ofthe original unit,then the rulesofoperation

must

be

suchthat the

number

which

represents the

measure

of

any

particu-larconcrete _length interms ofthe

new

unitshall be _{twice as large}

as the

number

which was

its

measure

in terms _{of the original} unit.

Very

little attentionseemsto_{have been given}tothe

_methodology

of systems of

measurement, and

I

do

not

know

whether

this

charac-teristic of all

our

systems of

measurement

has been formulated or

not,but itisevident

on

examinationof

any

system of

measurement

in actual use that it has _{these properties.}

The

_possession of this

ratio of the

numbers

_expressing the measures of

any two

concrete

lengths, for example, is _independent of the size of the unit with

which

they are measured. _{This consequence} is of course at once

obvious, for if

we

_change

the size of the

fundamental

unit

by any

factor,

by

hypothesis

we

change

the

measure

ofevery length

by

the

reciprocal of that factor,

and

so leave unaltered the ratio of the

measures

of

any two

lengths. This

means

that the ratio of the

lengths of

any two

particular objects has

an

absolute significance

independent ofthesize ofthe units. This

_may

be

_put

intothe

con-verse form,asisevident

on

aminute'sreflection. If

we

_{require that}

our system of

measurement

of

primary

kinds of_quantity in terms

of units of their

own

kindbesuch thatthe ratio of themeasures of

any two

concrete

examples

shall _{be independent} of the size of the

unit, then the measures of the concrete

examples must change

inversely as thesizeof theunit.

Besides

_primary

_{quantities, there}is another

group

of quantities

which

we

_may

call secondary quantities.

The

numerical

measures

of these are not obtained

_{by some}

_operation

which compares them

directly with another quantity of the

same

kind

which

is accepted

astheunit,butthe

method

of

measurement

is

more

complicated

and

roundabout. Quantitiesof thesecond kindare

measured by making

measurements

of certain quantities of thefirst kindassociatedwith

thequantity

under

consideration,

and

then

combining

the

measure-ments

of the associated

_primary

_{quantitiesaccording}tocertain rules

which

givea

number

thatisdefined as the

measure

of thesecondary quantity in question.

For

example, a velocity as ordinarily defined is _{a secondary quantity.}

We

obtain its

_{measure by measuring}

a length

and

the time occupied in traversing this length (both of

these being

primary

quantities),

and

dividingthe

number

measur-ingthe length

by

the

number

_measuring

thetime (or dividingthe

length

by

the time according to

our

shorthand

method

of

state-ment).

Now

there isa certain definite restriction

on

the rules of

opera-tion

which

we

are at liberty toset

_up

in _{defining secondary}

quan-tities.

We

make

the

same

requirementthat

we

did for

_primary

quan-tities, namely, that the ratio of the

numbers measuring any two

concrete

examples

of

a

secondary quantity shall be independent of

the size of the

fundamental

units used in

_making

the required

primary

measurements.

That

is, to say thatone substance istwice

travel-ling three times as rapidly as another, has absolute significance,

independent

of thesizeofthe

fundamental primary

units.

This requirementis_{not necessary}in orderto

make

measurement

itself possible.

Any

rules _{of operation} will serve as the basis of

a

system

of

_{measurement by which}

numbers

_may

be _assigned to

phenomena

insuch a

way

that theparticular aspectofthe

phenome-non on which

we

areconcentratingattentionisuniquelydefined

_by

the

number

in conjunction with the rules of operation.

But

the requirementthat the ratio be _constant, or

we

_may

saythe

require-ment

of the absolute significance of relative magnitude, is essential

to all the systems of

measurement

inscientific use. In particular, it is

an

absolute_requirementifthe

methods

of dimensional_analysis are to _{be applied} to the results of the measurements.

Dimensional

analysis cannot be applied to systems

which

do

not

meet

this

requirement,

and

accordingly

we

consider here only such systems.

Itis_particularlytobe noticedthat thelineofseparation

between

primary and

secondary quantities is not a

hard

and

fast one im-posed

by

natural conditions,butisto

a

large extent arbitrary,

and

depends on

the _particular set of rules of operation

which

we

find

convenient to

_adopt

in defining our system of

measurement.

For

instance, inour ordinary systemof mechanics, forceis _{a secondary}

quantity,

and

its

measure

is obtained

_by

multiplying

a

number

which measures

a

mass

and

the

number

which measures

an

accelera-tion _{(itselfa secondary quantity).}

But

physically, forceis_perfectly well

_adapted

tobe usedas a

_primary

_{quantity, since}

we know

what

we

mean

_by

sayingthatoneforceistwice _another,

and

the_physical

processes are

known

by which

force

may

be

measured

in terms of

units ofits

own

kind. Itisthe

same

_way

with velocities; it is

pos-sible to set

_up

a

physical procedure

by which

velocities

_may

be

added

together directly,

and which makes

it _possible to

measure

velocityintermsof units ofits

own

kind,

and

soto regardvelocity

asa

_primary

_quantity. Itis

_perhaps

questionable

whether

allkinds of physical quantity are

adapted

tobe treated,if itshould suit

our

convenience, as

_primary

quantities.

Thus

it is not at once obvious

whether

a_{physicalprocedure could be}set

_up

by which two

viscosi-tiescould be

compared

directlywith each other without

measuring

otherkinds_{of quantity.}

But

this _question is not essential to

our

progress, although of greatinterest in itself,

and need

not detain us.

The

facts are_simply