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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.


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