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Phys Lil

846

C612

THE LIBRARY

OF

THE UNIVERSITY

OF CALIFORNIA

LOS ANGELES

GIFT

FEB 2 1979

ELEMENTS OF DYNAMIC

ELEMENTS OF DYNAMIC

AN INTRODUCTION TO THE STUDY OF

MOTION AND REST

IN SOLID AND FLUID BODIES

W. K. CLIFFORD, F.R.S.

LATE PROFESSOR OF APPLIED MATHEMATICS AND MECHANICS AT

UNIVERSITY COLLEGE, LONDON.

PART I. KINEMATIC. BOOK IV. AND APPENDIX.

Hontron :

MACMILLAN AND CO.

AND NEW YOKE.

1887

[All Eights reserved.]

PRINTED BY C. J. CLAY, M.A. AND SONS,

AT THE UNIVERSITY PRESS.

lUDrarv

PREFACE.

I HAVE sufficiently explained in my letter to Mrs

Clifford (see " Mathematical Papers ") the reasons which

led me to accept the responsibility of editing the following

fragments. A few words as to the fragments themselves

may not be out of place here.

The first 56 pages are contained in 43 pages of MS.

These are carefully written out and paged, and in the

form in which they are left may be considered as nearly

representing that in which they would have been given

to the world by Clifford himself.

Pages 57 to 72 consist of detached portions of manu-

script written out in Clifford's usual careful manner, and

were evidently intended, after a further examination,

to take their places in his book. The remainder of

Appendix I. is printed here mainly with the view of

showing Clifford's work in its early stage. Thus (C) on

the " Top" is in its present form almost, if not quite,

unintelligible : most probably Clifford intended to discuss

the subject in connection with the "Kinetic analogy" of

Kirchhoff.

In Appendix II. I reprint the " Syllabus of Lectures

on Motion" from the "Papers" (pp. 516 524), chiefly

because it contains the article on Fourier's theorem

which was promised in the " Dynamic," p. 37 : and the

C. 6

VI PREFACE.

"Abstract of the Dynamic" because it passes with clear

and rapid touch over the subject as expounded in the

already published volume. The two "contents" (C) and

(D) put the reader in possession of what it was the

Author's intention to discuss had he lived to complete

his work.

I have not hesitated to extract from the Examination-

papers set by Clifford at University College a number of

questions, very characteristic of the Author, and to arrange

them as well as I could under the respective chapters :

in this course I have already met with warm approval.

I may mention that there should be added to my

" Bibliographical account" in the " Papers," a reference to

notes of a lecture on " Energy and Force," delivered by

Clifford before the Royal Institution on March 28, 1873.

Notes of this lecture, taken by Mr F. Pollock, and re-

vised by Mr J. F. Moulton, F.R.S., are published in

Nature, Vol. xxn. p. 123 (June 10, 1880). After con-

sulting with two or three mathematicians upon whose

judgment I could thoroughly rely, I have decided not to

insert these notes in the present volume. Clifford had

commenced an Index and had proceeded sufficiently far

to allow one to see on what lines he would have com-

pleted it : this task I have fulfilled on his lines.

R TUCKER.

CONTENTS.

BOOK IV. MASSES.

CHAPTER I. THE MASS-CENTRE.

PAGE

Density 1

MASS-CENTKE 2

Mass-Centre of Bod 3

Triangle and Tetrahedron 5

Quadrilaterals 6

Plane-faced Solids 8

Circular Arc and Sector 10

Eod of varying Density. Applications 11

Surface and Volume of Hemisphere 13

CHAPTER II. SECOND MOMENTS.

Plane Area 15

Parallel Axes. Swing-radius 16

Conjugate Axes. Pole of given Axis 17

Core of an Area 19

Swing-Conic 22

Poles and Polars 24

Application to the Null-conic 27

Equivalent Triad of Particles 28

Principal Axes .......... 29

Second Moments of a Solid 31

Swing-Ellipsoid 32

viii CONTENTS.

PAGE

Determination of the Pole of any Plane 34

Eelation of Pole to Swing-Quadric 35

Equivalent Tetrad of Particles 36

Second Moments in regard to an Axis 37

Ellipsoid of Gyration 38

Confocal Surfaces . 39

Principal Axes 42

Core of a Solid . 44

CHAPTER III. MOMENTUM.

Momentum of Translation- Velocity 48

Moment of Momentum 49

Rotor part of Momentum ........ 51

Momentum of Spins about Fixed Point 52

Momentum of Twist 55

APPENDIX I. (A)

Acceleration depending on Strain 57

Mass 58

Law of Combination . . . " 60

Law of Eeciprocity 60

Gravity 62

(B)

Electricity 64

Magnetism 65

Electric Currents 66

Law of Force 67

General Statement of the Laws of Motion 70

D'Alembert's Principle 71

(C)

Force 73

P)

The Rotation of a Rigid Body . 74

Moving Axes 75

Kinetic Energy 76

Top 77

CONTENTS. IX

(E)

PAGE

Energy of the resultant of a number of Motions .... 80

(F)

Momentum 81

APPENDIX II. (A)

Syllabus of Lectures on Motion 85

Fourier's Theorem 88

(B)

Abstract of ' Dynamic' 96

(0)

'CONTENTS' 101

(D)

ELEMENTS OF DYNAMIC (contents) 103

APPENDIX III.

Exercises 105

NOTES - 115

The references to the "Dynamic" Part I. are enclosed throughout in

square brackets [ ].

The following errata occur in that Volume :

p. 24, 1. 13, readej-e,,;

p. 102, 11 up, for - read = ;

p. 103, 1. 6, read an 2 ;

p. 131, 9 up, for J read ;

p. 132, 1. 3, for cos 6 read cot 6;

1. 6, insert - before h bis ;

1. 8, for X read h.

BOOK IV. MASSES.

CHAPTER I. THE MASS-CENTRE.

DENSITY.

WE have seen how to measure a change in the size or

volume of a body. When the size of a body is diminished,

it becomes more closely packed together, or more dense ;

when the size is increased, it becomes less dense. Suppose

that in a certain arbitrary state of the body we reckon its

density to be unity, then when it is compressed into

one-nth of the volume its density will be n times as great.

Or, if v is the volume of that which, at density 1, filled a

unit of volume, its density is now - . The density of a

body may be different in different parts ; the density of

the air, for example, diminishes as we go upwards. The

question then arises, how are we to compare different

portions of the same substance, so as to find out whether

they are of the same or different densities ? Given two

samples of air in bottles, or two samples of iron, one of

which has been hammered, how shall we compare their

densities ?

The answer is, that we must take equal volumes of the

two samples, and measure the quantity of stuff that there

is in each. For the two samples of air, we may put them

into perfectly flexible air-tight bags, so as not to fill the

bags ; then when these bags are held freely in the atmo-

sphere at the same level, the quantities of air are propor-

tional to the volumes they occupy. The two samples of

c. 1

2 DYNAMIC.

iron may be melted, and their volumes compared in that

state. For other substances the comparison by such

methods might be more difficult.

If a piece of stuff is of uniform density, the quantity of

stuff in it is the product of the volume and the density,

provided that the unit of quantity is taken to be that of

a unit of volume at unit density. The quantity of stuff in

a piece is called the mass or measure of that piece.

We shall give to the word mass a more extended

meaning when we come to consider the laws of motion*;

and shall then explain much easier methods of comparing

the masses of two pieces of the same stuff, as well as (in

the extended sense) of two pieces of different stuffs. For

the present, however, we shall suppose all the bodies

spoken of to be made of the same stuff, and we shall mean

by the mass of a given portion merely the quantity of that

stuff which it contains. All the results we shall get will

be applicable to the more extended meaning of the word.

When the density varies from point to point, the

density at any point is the mass which a unit of volume

would have if its density were uniformly equal to that at

the point-f*.

MASS-CENTRE.

If a particle of mass m be situated at a point p, the

vector m . op is called the mass-vector of the particle from

the origin o.

If a mass I be at a and a mass m

at &, a mass I + m at a point f such

that l.fa + m.fb =

shall be called the resultant of the two

masses.

Since we know that [p. 8]

I . oa + m . ob = (I + m) of,

it follows that the mass-vector of the resultant mass is

* [See below, p. 58.]

t [This sentence is hardly satisfactory, especially without any refer-

ence to the doctrine of limits.]

MASS-CENTRE. 3

equal to the sum of the mass-vectors of the components,

from any origin.

If there be a mass n at c, the resultant of I + m at f

and n at c will be called the resultant of I at a, m at b,

n at c, and so on for any number of particles. It follows

from the general theorem already proved that in all cases

the mass-vector of the resultant mass is the sum of the

mass-vectors of the components, from any origin.

The position of the resultant mass is called the centre

of mass or mass-centre of the given particles.

The moment of a particle in regard to any line or plane

is the product of the mass of the particle by its distance

from the line or plane.

The moment of the resultant mass is equal to the sum of

the moments of the components on any line or plane.

For let the origin o be taken in the given line or plane, oX

or oXZ; then the moment

of the particle I at p is equal

to the component of its mass-

vector I . op perpendicular to

the line or plane, namely,

I . mp. And since the mass-

vector of the resultant mass is

equal to the sum of the mass-

vectors of the component

masses, it follows that its

component perpendicular to any line or plane is the sum

of their components [p. 12].

MASS-CENTRE OF ROD.

If a mass be distributed uniformly along a straight line

ab, the mass-centre is at the middle point g of the line ;

for we may divide the line into pairs of particles equi-

distant from g, so that each pair has g for its mass-

centre.

We shall now verify that the moment of the resultant

mass is equal to the moment of the rod in regard to any

line through a perpendicular to ab.

12

4 DYNAMIC.

Let be, perpendicular to it, be equal in length to ab

multiplied by the mass of a unit

of length of it. Join ac, suppose

the length ab divided into small

portions of which mn is one, and

draw mk, nl perpendicular to ab

meeting ac in k, I. Then the

moment of mn in regard to a line

through a perpendicular to the

rod will lie between am multi-

plied by the mass of mn, and an multiplied by the same

mass. Now the moment of the mass of a unit of length

at m is mk, and at n is nl. Hence the moment of mn

lies between mn . mk and mn . nl. Thus the moment of

ab lies between two values which include the area abc and

which can be made as nearly equal as we like by increas-

ing the number of parts into which ab is divided. That is

to say, the moment of ab is equal to the area abc, namely

to \ ab . be = ab . gh,

where g is the middle point of ab. Now gh is the moment

of the mass of a unit of length at g ; therefore ab . gh

= moment of the mass of ab collected at g.

In the same way it appears that the moment of a por-

tion of the rod, such as mb, is equal to the area mbck

which stands over it.

In the case of a thin plate or lamina in the form of a

parallelogram, such that the masses of any two equal areas

of it are equal, the centre of mass is at the intersection of

the diagonals, which is also the intersection of the lines

joining the middle points of opposite sides (median lines).

For the area may be divided into thin strips by lines

parallel to one side, each of which has its mass-centre on a

median line.

And since a parallelepiped may be divided into such

thin plates by planes parallel to two of its faces, the

resultant masses of these will all lie in the straight line ab

joining the centres of those faces, if the density of the

solid be uniform (i.e. the masses of any two equal volumes

equal) ; and they will be distributed uniformly along this

TRIANGLE AND TETRAHEDRON. 5

line, since equal lengths of it will represent equal slices of

the solid ; therefore the centre of mass of the parallele-

piped is at the middle point g of ab.

TRIANGLE AND TETRAHEDRON.

In general, the moment of any plane lamina of uni-

form density (masses of equal

areas equal) about any line in its

plane is the volume of a solid

standing on the lamina, bounded

by lines through its boundary

perpendicular to its plane, and

by a plane drawn through the

straight line, such that the height

of every point of it is equal to

the moment about the line of

the mass of a unit of area situate directly under the

point. The proof of this is pre-

cisely similar to that of the case

of a uniform rod. The lamina is

to be divided into thin strips by

lines perpendicular to the given

line, and it is proved as in that

case that the moment of each of

these strips is the part of the

volume belonging to it.

If a plane lamina is such that all chords of it parallel

to a given direction are bisected by

a certain straight line, then the

centre of mass is in that straight

line. For the lamina may be di-

vided into thin strips like b'c, which

by cutting off small pieces at the

end, may be made into parallelo-

grams whose centre of mass is in

ad. The whole mass of the pieces 6 d

cut off may be made as small as we like by increasing

the number and diminishing the breadth of the strips.

Consequently the mass-centre must be in ad.

6

DYNAMIC.

In the case of the triangle, for example, the mass-

centre is in ad, and also in be; therefore it is in their

intersection g, which is also the mass-centre of three equal

particles at a, b, c, and one-third of the way from d to-

wards a.

So again, if all the sections of a solid by a series of

parallel planes have their mass-centres in the same straight

line, the mass-centre of the solid, supposed to be of uni-

form density, is in that line. For we may divide the

solid into thin slices by such parallel planes, which by

cutting off small pieces at their boundaries may be made

of uniform surface-density, and therefore have their re-

sultant masses on the given line. The mass of the pieces

cut off may be made as small as we like by increasing the

number and diminishing the thickness of the slices.

Thus, in the case of a tetrahedron, all plane sections

parallel to the face bed have their

mass-centres on the line aa, join-

ing the vertex a to the mass-

centre a of bed. Hence the mass-

centre of the tetrahedron is in aa.

Similarly it is in b(3, and therefore

in their intersection g. Hence it

coincides with the mass-centre of

four equal particles at a, b, c, d,

and is therefore one-quarter of

the way from a. towards a. It is

also the middle point of the three

lines which join the middle points

of opposite edges. [See p. 10.]

QUADRILATERALS.

To find the mass-centre of a trapezium, or quadrilateral

with two sides ad, be, par-

allel; we observe, first,

that it must be in the

line ef joining the middle

points of these sides, since

this line bisects all chords

parallel to them. Next,

QUADRILATERALS. 7

the trapezium being composed of the triangles adb, bdc,

its mass-centre must be in the line joining their mass-

centres, which are one-third way from e towards b and

from f towards d respectively ; and it must divide this

line in the inverse ratio of those triangles, that is, as

bf : ae, or say as b : a. Hence the middle third of e/"must

be divided in this ratio in g. The two parts being repre-

sented by 6 and a, the third of ef is represented by a + b ;

therefore ge is represented by (a + b) + b or a + 26, and

fg by a + (a + b) or 2a + b. Hence

eg : gf= a + 2b : 2a + b.

Make then ah equal to cb, and cJc equal to ad, then hk will

meet ef in the mass-centre g.

In a quadrilateral abed of any shape, let Tc, the inter-

section of the diagonals ac, bd,

be called the cross-centre, and

m, the middle of the line join-

ing their middle points e, f,

the mid-centre (mass-centre of

four equal particles at a, b, c,

d). The mass-centre fi of the

triangle acd or of three equal

particles at a, c, d, is in bm, so

that m/3 = %bm. Similarly 8

the mass-centre of abc is in dm so that raS = \dm. We

have to divide /8S in the inverse ratio of these triangles,

that is, as dk : kb. Hence the point required is where km

meets /38, and consequently

If we take ep = ke, and fq = kf, so that p, q are the

reflexions of k on the diagonals, g is mass-centre of the

triangle kpq. For it is on a line through e dividing @S

and therefore db in the ratio bk : kd, that is, on eq. Simi-

larly it is on pf.

8

DYNAMIC.

PLANE-FACED SOLIDS.

The triangle-faced pentacron is a solid made of two

tetrahedra with a common base.

The intersection Jc of the diagonal

ae and the diagonal plane bed is the

cross-centre. Let p be the reflexion

of k on ae, so that pe = ok, and f the

mass-centre of bed ; then g the mass-

centre of the solid is one-quarter

way from f to p. For the mass-

centres a and e of the tetrahedra

ebcd, abed are one-quarter way from

f to e and a respectively, and fg

divides ae, and therefore ea, in the

ratio ek : ka. Consequently it passes through p. If m

be the mid-centre (mass-centre of equal particles at

a, b, c, d, e ; it is f way from f to h the middle of ae)

we know that mi = %am, me = \em\ hence mg = \km,

or the mass-centre of the solid is in the prolongation of

the line joining the cross- and mid-centres, at a distance

from the latter equal to one- quarter of the distance be-

tween them.

The octahedron abcdefis one form of triangle-faced hexa-

cron and may be regarded as made of

two pyramids abcef, dbcef standing

on a common skew quadrilateral base

beef. There are three such quadrila-

terals, the other two being aedb, acdf.

It is to be understood that,in general,

no two of the diagonals ad, be, cf

intersect, so that no one of these

quadrilaterals is plane. But the

middle points of the sides of a skew

quadrilateral are always in one plane ;

for (e.g.) the line joining the middles of bf,fe, and the line

joining the middles of be, ce, are both parallel to be, and

any two parallel lines are in one plane. To each of the

three quadrilaterals there is such a plane, and the in-

tersection k of these planes is called the cross-centre. The

PLANE-FACED SOLIDS. 9

mid-centre m, or mass-centre of equal particles placed at

the vertices, is the mass-centre of the middle points of

the three diagonals. Now the solid is the sum of the

four tetrahedra adef, adfb, adbc, adce, and therefore its

mass-centre g is in the plane containing their mass-centres.

Now the mass-centre of adef, say x, being also the mass-

centre of four equal particles at a, d, e,f, is on the line

joining the mid-centre with the middle point p of be,

so that asm = ^mp. Hence the plane through the mass-

centres x, y, z, w of the four tetrahedra just mentioned,

is parallel to the plane through the middle points pqrs of

be, ce, ef, fb, and at half the distance from m of that

plane. Each of the diagonals gives rise to such a division

of the solid into tetrahedra, and it follows that the mass-

centre g lies on each of three planes parallel to the

three planes which intersect in the cross-centre k and

at half the distance from m. Hence g is on the line km,

so that

A particular case of this solid is the frustum of a tetra-

hedron, one vertex of which is v

cut off by a plane section. It

occurs when the faces afe, abf

are in one plane, as also bdf,

bed, and ced, cae. We may

either count (as is here done)

af, bd, ce for edges of the solid,

and ad, be, cf for diagonals ; or

else we may take ad, be, cf for

edges, and af, bd, ce for diago-

nals. In the former case the

cross-centre is the intersection

of planes through the middle points of the quadrilaterals

beef, cfad, adbe ; in the latter case the quadrilaterals are

bdce, ceaf, afbd. The cross-centre is of course the same

point in either case, so that these six planes intersect in a

point k. The position of g is given as in the general

case by

mg = ^ km.

10 DYNAMIC.

The other form of triangle-faced hexacron is shewn in

the figure. Each of the vertices e

e and f has five edges through

it, b and c have four, a and d

three ; while in the octahedron

every vertex has four edges

through it. The construction

of the cross-centre is not quite

so simple in this case. Let p

be a point one-fourth of the way from the middle of ac to

the middle of cd, and q a point one-fourth of the way from

the middle of bd to the middle of ab. Through p draw a

plane parallel to cef, and through q a plane parallel to bef.

The intersection of these planes with a plane through

the middle points of ba, ad, dc is the cross-centre k. If m

be the mid-centre and g the mass-centre, then as before

we have mg = \ km.

To prove this we observe that the solid is the sum of

the four tetrahedra efab, bdec, bdcf, bdfe, and that their

mass-centres are on straight lines through the mid-centre

and the middle points of cd, of, ae, ac respectively, at

half the distance of these latter from m. The middles of

af, ae, ac are in a plane parallel to efc at half its distance

from a. The mass-centre of these points and the middle

of cd is one-fourth the way from a point in this plane to

the middle of cd. It is therefore in a plane parallel to efc

through p, which is one-fourth the way from the middle of

ac to the middle of cd. Similarly by dividing the solid

into the tetrahedra efab, efbc, efcd, we may shew that k is

in a plane through the middles of ba, ad, dc.

The method of the preceding five paragraphs, the

useful names of the mid-centre and cross-centre, and the

theorem for the tetrahedral frustum are due to Sylvester*.

CIRCULAR ARC AND SECTOR.

The moment of a circular arc in regard to any line

through the centre is equal to its projection multiplied by

* Phil. Mag. [Vol. xxvi., pp. 167183 (1863). See Math. Papers,

p. 409, or Proc. of Land. Math. Soc. Vol. ix. p. 28.]

CIRCULAR ARC AND SECTOR. 11

the radius. Let pq be a tan-

gent at r, or perpendicular on

it from the centre, ra, n, I the

projections of q, r, p. Then

the moment of pq is

pq . rn = ml . or,

since

ml : pq = rn : or = sin rox.

Thus the moment of every piece of straight line is equal

to its projection multiplied by its perpendicular distance

from the origin. If we draw a polygon circumscribing the

circular arc, the distance of all its sides from the origin is

equal to the radius of the circle ; and therefore its moment

is the radius multiplied by its projection. Such a polygon

may be made to approximate as nearly as we like to the

circle by increasing the number of sides and diminishing

their length ; therefore the same thing is true for the

circle.

Taking now ox parallel to the chord of the arc, we find

the distance of its mass-centre from o to be

radius x chord db : length of arc,

or if the angle aob = 20, radius = a, then this distance is

asin# : 6.

A circular sector may be approximately divided into

small triangles whose vertex is at o and whose mass-

centres are distant fa from o. Hence it is equivalent to a

uniform arc of two-thirds the radius, and the distance from

o of the mass-centre is 2a sin 6 : 30. Thus in the case of

a semi-circle, 6 = ^ TT, distance of mass-centre from o

ROD OF VARYING DENSITY. APPLICATIONS.

A rod whose density varies as the distance from one end

is equivalent to a uniform triangle with its base bisected

by the other end, and therefore its mass-centre is of

the length from the other end. If the density varies as

the square of the distance from one end, the rod may be

regarded as a uniform tetrahedron whose base has its mass-

12 DYNAMIC.

centre at the other end ; consequently the mass-centre is

one-fourth of the length from the other end. Generally,

suppose the density to vary as the

kih power of the distance x from ^i x

one end, a. Then the mass of a * *

small length See, one point of which

is at distance x from a, would be x h $x if the density in Bx

were uniformly what it is at the distance x. Thus ^x k 8x

is an approximation to the mass of the rod, which can be

made as close as we like by diminishing the Bx and in-

creasing their number. Hence the mass of the rod

= x k dx = a k+i : k + l.

J o

Similarly the moment of 8x about a is approximately

x . atSx or x k+l 8x )

and therefore the moment of the rod is

Therefore distance of mass-centre

= a(k + l) :

or, from the other end,

= a : k+2.

Thus the two examples given are cases of this general

rule : In a rod whose density varies as the kih power of

the distance from one end, the mass-centre is one (k + 2)th

part of the length from the other end.

846

C612

THE LIBRARY

OF

THE UNIVERSITY

OF CALIFORNIA

LOS ANGELES

GIFT

FEB 2 1979

ELEMENTS OF DYNAMIC

ELEMENTS OF DYNAMIC

AN INTRODUCTION TO THE STUDY OF

MOTION AND REST

IN SOLID AND FLUID BODIES

W. K. CLIFFORD, F.R.S.

LATE PROFESSOR OF APPLIED MATHEMATICS AND MECHANICS AT

UNIVERSITY COLLEGE, LONDON.

PART I. KINEMATIC. BOOK IV. AND APPENDIX.

Hontron :

MACMILLAN AND CO.

AND NEW YOKE.

1887

[All Eights reserved.]

PRINTED BY C. J. CLAY, M.A. AND SONS,

AT THE UNIVERSITY PRESS.

lUDrarv

PREFACE.

I HAVE sufficiently explained in my letter to Mrs

Clifford (see " Mathematical Papers ") the reasons which

led me to accept the responsibility of editing the following

fragments. A few words as to the fragments themselves

may not be out of place here.

The first 56 pages are contained in 43 pages of MS.

These are carefully written out and paged, and in the

form in which they are left may be considered as nearly

representing that in which they would have been given

to the world by Clifford himself.

Pages 57 to 72 consist of detached portions of manu-

script written out in Clifford's usual careful manner, and

were evidently intended, after a further examination,

to take their places in his book. The remainder of

Appendix I. is printed here mainly with the view of

showing Clifford's work in its early stage. Thus (C) on

the " Top" is in its present form almost, if not quite,

unintelligible : most probably Clifford intended to discuss

the subject in connection with the "Kinetic analogy" of

Kirchhoff.

In Appendix II. I reprint the " Syllabus of Lectures

on Motion" from the "Papers" (pp. 516 524), chiefly

because it contains the article on Fourier's theorem

which was promised in the " Dynamic," p. 37 : and the

C. 6

VI PREFACE.

"Abstract of the Dynamic" because it passes with clear

and rapid touch over the subject as expounded in the

already published volume. The two "contents" (C) and

(D) put the reader in possession of what it was the

Author's intention to discuss had he lived to complete

his work.

I have not hesitated to extract from the Examination-

papers set by Clifford at University College a number of

questions, very characteristic of the Author, and to arrange

them as well as I could under the respective chapters :

in this course I have already met with warm approval.

I may mention that there should be added to my

" Bibliographical account" in the " Papers," a reference to

notes of a lecture on " Energy and Force," delivered by

Clifford before the Royal Institution on March 28, 1873.

Notes of this lecture, taken by Mr F. Pollock, and re-

vised by Mr J. F. Moulton, F.R.S., are published in

Nature, Vol. xxn. p. 123 (June 10, 1880). After con-

sulting with two or three mathematicians upon whose

judgment I could thoroughly rely, I have decided not to

insert these notes in the present volume. Clifford had

commenced an Index and had proceeded sufficiently far

to allow one to see on what lines he would have com-

pleted it : this task I have fulfilled on his lines.

R TUCKER.

CONTENTS.

BOOK IV. MASSES.

CHAPTER I. THE MASS-CENTRE.

PAGE

Density 1

MASS-CENTKE 2

Mass-Centre of Bod 3

Triangle and Tetrahedron 5

Quadrilaterals 6

Plane-faced Solids 8

Circular Arc and Sector 10

Eod of varying Density. Applications 11

Surface and Volume of Hemisphere 13

CHAPTER II. SECOND MOMENTS.

Plane Area 15

Parallel Axes. Swing-radius 16

Conjugate Axes. Pole of given Axis 17

Core of an Area 19

Swing-Conic 22

Poles and Polars 24

Application to the Null-conic 27

Equivalent Triad of Particles 28

Principal Axes .......... 29

Second Moments of a Solid 31

Swing-Ellipsoid 32

viii CONTENTS.

PAGE

Determination of the Pole of any Plane 34

Eelation of Pole to Swing-Quadric 35

Equivalent Tetrad of Particles 36

Second Moments in regard to an Axis 37

Ellipsoid of Gyration 38

Confocal Surfaces . 39

Principal Axes 42

Core of a Solid . 44

CHAPTER III. MOMENTUM.

Momentum of Translation- Velocity 48

Moment of Momentum 49

Rotor part of Momentum ........ 51

Momentum of Spins about Fixed Point 52

Momentum of Twist 55

APPENDIX I. (A)

Acceleration depending on Strain 57

Mass 58

Law of Combination . . . " 60

Law of Eeciprocity 60

Gravity 62

(B)

Electricity 64

Magnetism 65

Electric Currents 66

Law of Force 67

General Statement of the Laws of Motion 70

D'Alembert's Principle 71

(C)

Force 73

P)

The Rotation of a Rigid Body . 74

Moving Axes 75

Kinetic Energy 76

Top 77

CONTENTS. IX

(E)

PAGE

Energy of the resultant of a number of Motions .... 80

(F)

Momentum 81

APPENDIX II. (A)

Syllabus of Lectures on Motion 85

Fourier's Theorem 88

(B)

Abstract of ' Dynamic' 96

(0)

'CONTENTS' 101

(D)

ELEMENTS OF DYNAMIC (contents) 103

APPENDIX III.

Exercises 105

NOTES - 115

The references to the "Dynamic" Part I. are enclosed throughout in

square brackets [ ].

The following errata occur in that Volume :

p. 24, 1. 13, readej-e,,;

p. 102, 11 up, for - read = ;

p. 103, 1. 6, read an 2 ;

p. 131, 9 up, for J read ;

p. 132, 1. 3, for cos 6 read cot 6;

1. 6, insert - before h bis ;

1. 8, for X read h.

BOOK IV. MASSES.

CHAPTER I. THE MASS-CENTRE.

DENSITY.

WE have seen how to measure a change in the size or

volume of a body. When the size of a body is diminished,

it becomes more closely packed together, or more dense ;

when the size is increased, it becomes less dense. Suppose

that in a certain arbitrary state of the body we reckon its

density to be unity, then when it is compressed into

one-nth of the volume its density will be n times as great.

Or, if v is the volume of that which, at density 1, filled a

unit of volume, its density is now - . The density of a

body may be different in different parts ; the density of

the air, for example, diminishes as we go upwards. The

question then arises, how are we to compare different

portions of the same substance, so as to find out whether

they are of the same or different densities ? Given two

samples of air in bottles, or two samples of iron, one of

which has been hammered, how shall we compare their

densities ?

The answer is, that we must take equal volumes of the

two samples, and measure the quantity of stuff that there

is in each. For the two samples of air, we may put them

into perfectly flexible air-tight bags, so as not to fill the

bags ; then when these bags are held freely in the atmo-

sphere at the same level, the quantities of air are propor-

tional to the volumes they occupy. The two samples of

c. 1

2 DYNAMIC.

iron may be melted, and their volumes compared in that

state. For other substances the comparison by such

methods might be more difficult.

If a piece of stuff is of uniform density, the quantity of

stuff in it is the product of the volume and the density,

provided that the unit of quantity is taken to be that of

a unit of volume at unit density. The quantity of stuff in

a piece is called the mass or measure of that piece.

We shall give to the word mass a more extended

meaning when we come to consider the laws of motion*;

and shall then explain much easier methods of comparing

the masses of two pieces of the same stuff, as well as (in

the extended sense) of two pieces of different stuffs. For

the present, however, we shall suppose all the bodies

spoken of to be made of the same stuff, and we shall mean

by the mass of a given portion merely the quantity of that

stuff which it contains. All the results we shall get will

be applicable to the more extended meaning of the word.

When the density varies from point to point, the

density at any point is the mass which a unit of volume

would have if its density were uniformly equal to that at

the point-f*.

MASS-CENTRE.

If a particle of mass m be situated at a point p, the

vector m . op is called the mass-vector of the particle from

the origin o.

If a mass I be at a and a mass m

at &, a mass I + m at a point f such

that l.fa + m.fb =

shall be called the resultant of the two

masses.

Since we know that [p. 8]

I . oa + m . ob = (I + m) of,

it follows that the mass-vector of the resultant mass is

* [See below, p. 58.]

t [This sentence is hardly satisfactory, especially without any refer-

ence to the doctrine of limits.]

MASS-CENTRE. 3

equal to the sum of the mass-vectors of the components,

from any origin.

If there be a mass n at c, the resultant of I + m at f

and n at c will be called the resultant of I at a, m at b,

n at c, and so on for any number of particles. It follows

from the general theorem already proved that in all cases

the mass-vector of the resultant mass is the sum of the

mass-vectors of the components, from any origin.

The position of the resultant mass is called the centre

of mass or mass-centre of the given particles.

The moment of a particle in regard to any line or plane

is the product of the mass of the particle by its distance

from the line or plane.

The moment of the resultant mass is equal to the sum of

the moments of the components on any line or plane.

For let the origin o be taken in the given line or plane, oX

or oXZ; then the moment

of the particle I at p is equal

to the component of its mass-

vector I . op perpendicular to

the line or plane, namely,

I . mp. And since the mass-

vector of the resultant mass is

equal to the sum of the mass-

vectors of the component

masses, it follows that its

component perpendicular to any line or plane is the sum

of their components [p. 12].

MASS-CENTRE OF ROD.

If a mass be distributed uniformly along a straight line

ab, the mass-centre is at the middle point g of the line ;

for we may divide the line into pairs of particles equi-

distant from g, so that each pair has g for its mass-

centre.

We shall now verify that the moment of the resultant

mass is equal to the moment of the rod in regard to any

line through a perpendicular to ab.

12

4 DYNAMIC.

Let be, perpendicular to it, be equal in length to ab

multiplied by the mass of a unit

of length of it. Join ac, suppose

the length ab divided into small

portions of which mn is one, and

draw mk, nl perpendicular to ab

meeting ac in k, I. Then the

moment of mn in regard to a line

through a perpendicular to the

rod will lie between am multi-

plied by the mass of mn, and an multiplied by the same

mass. Now the moment of the mass of a unit of length

at m is mk, and at n is nl. Hence the moment of mn

lies between mn . mk and mn . nl. Thus the moment of

ab lies between two values which include the area abc and

which can be made as nearly equal as we like by increas-

ing the number of parts into which ab is divided. That is

to say, the moment of ab is equal to the area abc, namely

to \ ab . be = ab . gh,

where g is the middle point of ab. Now gh is the moment

of the mass of a unit of length at g ; therefore ab . gh

= moment of the mass of ab collected at g.

In the same way it appears that the moment of a por-

tion of the rod, such as mb, is equal to the area mbck

which stands over it.

In the case of a thin plate or lamina in the form of a

parallelogram, such that the masses of any two equal areas

of it are equal, the centre of mass is at the intersection of

the diagonals, which is also the intersection of the lines

joining the middle points of opposite sides (median lines).

For the area may be divided into thin strips by lines

parallel to one side, each of which has its mass-centre on a

median line.

And since a parallelepiped may be divided into such

thin plates by planes parallel to two of its faces, the

resultant masses of these will all lie in the straight line ab

joining the centres of those faces, if the density of the

solid be uniform (i.e. the masses of any two equal volumes

equal) ; and they will be distributed uniformly along this

TRIANGLE AND TETRAHEDRON. 5

line, since equal lengths of it will represent equal slices of

the solid ; therefore the centre of mass of the parallele-

piped is at the middle point g of ab.

TRIANGLE AND TETRAHEDRON.

In general, the moment of any plane lamina of uni-

form density (masses of equal

areas equal) about any line in its

plane is the volume of a solid

standing on the lamina, bounded

by lines through its boundary

perpendicular to its plane, and

by a plane drawn through the

straight line, such that the height

of every point of it is equal to

the moment about the line of

the mass of a unit of area situate directly under the

point. The proof of this is pre-

cisely similar to that of the case

of a uniform rod. The lamina is

to be divided into thin strips by

lines perpendicular to the given

line, and it is proved as in that

case that the moment of each of

these strips is the part of the

volume belonging to it.

If a plane lamina is such that all chords of it parallel

to a given direction are bisected by

a certain straight line, then the

centre of mass is in that straight

line. For the lamina may be di-

vided into thin strips like b'c, which

by cutting off small pieces at the

end, may be made into parallelo-

grams whose centre of mass is in

ad. The whole mass of the pieces 6 d

cut off may be made as small as we like by increasing

the number and diminishing the breadth of the strips.

Consequently the mass-centre must be in ad.

6

DYNAMIC.

In the case of the triangle, for example, the mass-

centre is in ad, and also in be; therefore it is in their

intersection g, which is also the mass-centre of three equal

particles at a, b, c, and one-third of the way from d to-

wards a.

So again, if all the sections of a solid by a series of

parallel planes have their mass-centres in the same straight

line, the mass-centre of the solid, supposed to be of uni-

form density, is in that line. For we may divide the

solid into thin slices by such parallel planes, which by

cutting off small pieces at their boundaries may be made

of uniform surface-density, and therefore have their re-

sultant masses on the given line. The mass of the pieces

cut off may be made as small as we like by increasing the

number and diminishing the thickness of the slices.

Thus, in the case of a tetrahedron, all plane sections

parallel to the face bed have their

mass-centres on the line aa, join-

ing the vertex a to the mass-

centre a of bed. Hence the mass-

centre of the tetrahedron is in aa.

Similarly it is in b(3, and therefore

in their intersection g. Hence it

coincides with the mass-centre of

four equal particles at a, b, c, d,

and is therefore one-quarter of

the way from a. towards a. It is

also the middle point of the three

lines which join the middle points

of opposite edges. [See p. 10.]

QUADRILATERALS.

To find the mass-centre of a trapezium, or quadrilateral

with two sides ad, be, par-

allel; we observe, first,

that it must be in the

line ef joining the middle

points of these sides, since

this line bisects all chords

parallel to them. Next,

QUADRILATERALS. 7

the trapezium being composed of the triangles adb, bdc,

its mass-centre must be in the line joining their mass-

centres, which are one-third way from e towards b and

from f towards d respectively ; and it must divide this

line in the inverse ratio of those triangles, that is, as

bf : ae, or say as b : a. Hence the middle third of e/"must

be divided in this ratio in g. The two parts being repre-

sented by 6 and a, the third of ef is represented by a + b ;

therefore ge is represented by (a + b) + b or a + 26, and

fg by a + (a + b) or 2a + b. Hence

eg : gf= a + 2b : 2a + b.

Make then ah equal to cb, and cJc equal to ad, then hk will

meet ef in the mass-centre g.

In a quadrilateral abed of any shape, let Tc, the inter-

section of the diagonals ac, bd,

be called the cross-centre, and

m, the middle of the line join-

ing their middle points e, f,

the mid-centre (mass-centre of

four equal particles at a, b, c,

d). The mass-centre fi of the

triangle acd or of three equal

particles at a, c, d, is in bm, so

that m/3 = %bm. Similarly 8

the mass-centre of abc is in dm so that raS = \dm. We

have to divide /8S in the inverse ratio of these triangles,

that is, as dk : kb. Hence the point required is where km

meets /38, and consequently

If we take ep = ke, and fq = kf, so that p, q are the

reflexions of k on the diagonals, g is mass-centre of the

triangle kpq. For it is on a line through e dividing @S

and therefore db in the ratio bk : kd, that is, on eq. Simi-

larly it is on pf.

8

DYNAMIC.

PLANE-FACED SOLIDS.

The triangle-faced pentacron is a solid made of two

tetrahedra with a common base.

The intersection Jc of the diagonal

ae and the diagonal plane bed is the

cross-centre. Let p be the reflexion

of k on ae, so that pe = ok, and f the

mass-centre of bed ; then g the mass-

centre of the solid is one-quarter

way from f to p. For the mass-

centres a and e of the tetrahedra

ebcd, abed are one-quarter way from

f to e and a respectively, and fg

divides ae, and therefore ea, in the

ratio ek : ka. Consequently it passes through p. If m

be the mid-centre (mass-centre of equal particles at

a, b, c, d, e ; it is f way from f to h the middle of ae)

we know that mi = %am, me = \em\ hence mg = \km,

or the mass-centre of the solid is in the prolongation of

the line joining the cross- and mid-centres, at a distance

from the latter equal to one- quarter of the distance be-

tween them.

The octahedron abcdefis one form of triangle-faced hexa-

cron and may be regarded as made of

two pyramids abcef, dbcef standing

on a common skew quadrilateral base

beef. There are three such quadrila-

terals, the other two being aedb, acdf.

It is to be understood that,in general,

no two of the diagonals ad, be, cf

intersect, so that no one of these

quadrilaterals is plane. But the

middle points of the sides of a skew

quadrilateral are always in one plane ;

for (e.g.) the line joining the middles of bf,fe, and the line

joining the middles of be, ce, are both parallel to be, and

any two parallel lines are in one plane. To each of the

three quadrilaterals there is such a plane, and the in-

tersection k of these planes is called the cross-centre. The

PLANE-FACED SOLIDS. 9

mid-centre m, or mass-centre of equal particles placed at

the vertices, is the mass-centre of the middle points of

the three diagonals. Now the solid is the sum of the

four tetrahedra adef, adfb, adbc, adce, and therefore its

mass-centre g is in the plane containing their mass-centres.

Now the mass-centre of adef, say x, being also the mass-

centre of four equal particles at a, d, e,f, is on the line

joining the mid-centre with the middle point p of be,

so that asm = ^mp. Hence the plane through the mass-

centres x, y, z, w of the four tetrahedra just mentioned,

is parallel to the plane through the middle points pqrs of

be, ce, ef, fb, and at half the distance from m of that

plane. Each of the diagonals gives rise to such a division

of the solid into tetrahedra, and it follows that the mass-

centre g lies on each of three planes parallel to the

three planes which intersect in the cross-centre k and

at half the distance from m. Hence g is on the line km,

so that

A particular case of this solid is the frustum of a tetra-

hedron, one vertex of which is v

cut off by a plane section. It

occurs when the faces afe, abf

are in one plane, as also bdf,

bed, and ced, cae. We may

either count (as is here done)

af, bd, ce for edges of the solid,

and ad, be, cf for diagonals ; or

else we may take ad, be, cf for

edges, and af, bd, ce for diago-

nals. In the former case the

cross-centre is the intersection

of planes through the middle points of the quadrilaterals

beef, cfad, adbe ; in the latter case the quadrilaterals are

bdce, ceaf, afbd. The cross-centre is of course the same

point in either case, so that these six planes intersect in a

point k. The position of g is given as in the general

case by

mg = ^ km.

10 DYNAMIC.

The other form of triangle-faced hexacron is shewn in

the figure. Each of the vertices e

e and f has five edges through

it, b and c have four, a and d

three ; while in the octahedron

every vertex has four edges

through it. The construction

of the cross-centre is not quite

so simple in this case. Let p

be a point one-fourth of the way from the middle of ac to

the middle of cd, and q a point one-fourth of the way from

the middle of bd to the middle of ab. Through p draw a

plane parallel to cef, and through q a plane parallel to bef.

The intersection of these planes with a plane through

the middle points of ba, ad, dc is the cross-centre k. If m

be the mid-centre and g the mass-centre, then as before

we have mg = \ km.

To prove this we observe that the solid is the sum of

the four tetrahedra efab, bdec, bdcf, bdfe, and that their

mass-centres are on straight lines through the mid-centre

and the middle points of cd, of, ae, ac respectively, at

half the distance of these latter from m. The middles of

af, ae, ac are in a plane parallel to efc at half its distance

from a. The mass-centre of these points and the middle

of cd is one-fourth the way from a point in this plane to

the middle of cd. It is therefore in a plane parallel to efc

through p, which is one-fourth the way from the middle of

ac to the middle of cd. Similarly by dividing the solid

into the tetrahedra efab, efbc, efcd, we may shew that k is

in a plane through the middles of ba, ad, dc.

The method of the preceding five paragraphs, the

useful names of the mid-centre and cross-centre, and the

theorem for the tetrahedral frustum are due to Sylvester*.

CIRCULAR ARC AND SECTOR.

The moment of a circular arc in regard to any line

through the centre is equal to its projection multiplied by

* Phil. Mag. [Vol. xxvi., pp. 167183 (1863). See Math. Papers,

p. 409, or Proc. of Land. Math. Soc. Vol. ix. p. 28.]

CIRCULAR ARC AND SECTOR. 11

the radius. Let pq be a tan-

gent at r, or perpendicular on

it from the centre, ra, n, I the

projections of q, r, p. Then

the moment of pq is

pq . rn = ml . or,

since

ml : pq = rn : or = sin rox.

Thus the moment of every piece of straight line is equal

to its projection multiplied by its perpendicular distance

from the origin. If we draw a polygon circumscribing the

circular arc, the distance of all its sides from the origin is

equal to the radius of the circle ; and therefore its moment

is the radius multiplied by its projection. Such a polygon

may be made to approximate as nearly as we like to the

circle by increasing the number of sides and diminishing

their length ; therefore the same thing is true for the

circle.

Taking now ox parallel to the chord of the arc, we find

the distance of its mass-centre from o to be

radius x chord db : length of arc,

or if the angle aob = 20, radius = a, then this distance is

asin# : 6.

A circular sector may be approximately divided into

small triangles whose vertex is at o and whose mass-

centres are distant fa from o. Hence it is equivalent to a

uniform arc of two-thirds the radius, and the distance from

o of the mass-centre is 2a sin 6 : 30. Thus in the case of

a semi-circle, 6 = ^ TT, distance of mass-centre from o

ROD OF VARYING DENSITY. APPLICATIONS.

A rod whose density varies as the distance from one end

is equivalent to a uniform triangle with its base bisected

by the other end, and therefore its mass-centre is of

the length from the other end. If the density varies as

the square of the distance from one end, the rod may be

regarded as a uniform tetrahedron whose base has its mass-

12 DYNAMIC.

centre at the other end ; consequently the mass-centre is

one-fourth of the length from the other end. Generally,

suppose the density to vary as the

kih power of the distance x from ^i x

one end, a. Then the mass of a * *

small length See, one point of which

is at distance x from a, would be x h $x if the density in Bx

were uniformly what it is at the distance x. Thus ^x k 8x

is an approximation to the mass of the rod, which can be

made as close as we like by diminishing the Bx and in-

creasing their number. Hence the mass of the rod

= x k dx = a k+i : k + l.

J o

Similarly the moment of 8x about a is approximately

x . atSx or x k+l 8x )

and therefore the moment of the rod is

Therefore distance of mass-centre

= a(k + l) :

or, from the other end,

= a : k+2.

Thus the two examples given are cases of this general

rule : In a rod whose density varies as the kih power of

the distance from one end, the mass-centre is one (k + 2)th

part of the length from the other end.