William Kingdon Clifford.

# Elements of dynamic; an introduction to the study of motion and rest in solid and fluid bodies (Volume 2) online

. (page 3 of 9)
Font size pole of the opposite side;
such a triangle is called self-

conjugate*. Let xg meet yz in u, then xg . gu is the
square of the semi-diameter along xu of the swing-conic
at the mass-centre g. If a line through g parallel to yz
meet xz in v and yw, parallel to g.v, in w, then v is the
pole of yw and consequently wg . gv is the square of the
semi-diameter along wv. Thus if x, y, z and g are
known, we can find the lengths and position of two conju-
gate semi-diameters of the swing-conic ; that is, the swing-
conic is determined.

Now when we know the mass a of an area and the
swing-conic at the mass-centre, the second moment in
regard to every line in its plane is determined. For it is
a (A 2 + a 2 ) sin' 2 6, where a is the conjugate semi-diameter,
h the distance from the centre measured along it, and
the angle it makes with the given line.

If three particles be placed at xyz, their masses being
in such proportion that their mass-centre is g, they will
constitute a system (regarded as an area shrunk up into
three points) which obviously has xyz for a self-conjugate
triangle, and therefore the same swing-conic at g as the
given area. If we now make the sum of their masses = a,
their second moment about any line in the plane is the
same as that of the given area. Thus so far as second
moments are concerned, an area may be replaced by three
particles, forming a self-conjugate triangle, of the same
resultant mass as the given area.

PRINCIPAL AXES.

The axes of the swing-conic at any point are called
the principal axes of the area at that point. The principal
axes at the mass-centre are often called for shortness
principal axes of the area.

Let oX, o Fbe the principal
axes at the mass-centre o, and
let b, a be their swing -radii.
of a line, the perpendicular
from the origin on which is of

* [Self-conjugate with regard to the area, it is not self-conjugate
with regard to swing-conic.]

DYNAMIC.

length p and inclination a to the axis oX, is

]<? = a 2 cos 8 a + 6 2 sin 2 a + p 2 .
Consequently p z = (k 2 a 2 ) cos 2 a + (k z Z> 2 ) sin 2 a ;

whereby it appears that this line touches a conic the
squared semi-axes of which are k* a 2 , k 2 b 2 .

Suppose that a is greater than b ; then the swing-conic
is an ellipse whose foci are on oX &t distances \/(a 2 6 2 )
from o on either side. The conic whose squared semi-
axes are k 2 a~, k 2 b* has its foci on oY at the same
distances from o. Since the squared semi-axes of the
null-conic are a 2 , b 2 , it also has its foci on oY at this
distance. Hence all the lines whose swing-radius is equal
to a given quantity k touch a conic confocal with the null-
conic.

Let Sj, Aj be foci of the swing-conic, s, h of the null-
conic and of all the conies of constant swing-radius. When
k is greater than a, the conic is an ellipse; when it is
between a and b, a hyperbola; when less than b, invisible.
Through every point p pass one
ellipse of the series, for which
the sum of the focal distances is
sp + hp; and one hyperbola, for
which their difference is sp ~ hp.
Hence every ellipse cuts every
hyperbola, but no two ellipses
or hyperbola intersect. The
curves cut at right angles, be-
cause their respective tangents
bisect internally and externally
the angle sph. To pass from
any ellipse to a larger one is to
increase the value of k.

The principal axes at any point are tangents to the
ellipse and hyperbola confocal to the null-conic which pass
through that point. For clearly one principal axis has the
greatest swing-radius, and the other the least, of all axes
passing through the point. Now consider the tangent to
an ellipse of the system at a point p. Its swing-radius is

SECOND MOMENTS OF A SOLID. 31

the quantity k belonging to that ellipse. If we turn the
line round p a little, it will cut the ellipse ; thus the
curve of the series which it touches is an ellipse inside
the given one, which will therefore belong to a less value
of Jc. Thus by turning the tangent round, we diminish
its swing-radius ; or the tangent to the ellipse is the line
of greatest swing-radius through p ; that is, it is one of
the principal axes. Similarly the tangent to the hyper-
bola is the line of least swing-radius, and therefore the
other principal axis.

SECOND MOMENTS OF A SOLID.

The preceding theory may now be very readily extended
to three dimensions.

If a body be altered by multiplying its density at every
point by the distance from a given plane, it is said to be
loaded from that plane. Distances on one side of the
plane being reckoned positive, those on the other side are
reckoned negative ; thus the density of the body is altered
in sign on one side of the plane.

The first moment of the loaded body in regard to the
plane is called the second moment of the unloaded body
in regard to the plane. If the body be divided into
particles, the mass of one of which is 8m, the second
moment is what 2# 2 6'ra indefinitely approximates to with
diminishing 8m; that is, it is ]x*dm; where x is the
distance of 8m from the plane.

The mass-centre of the loaded body is called the pole
of the plane in regard to the unloaded body.

If one plane pass through the pole of another, the
second plane passes through the pole of the first. For in
that case fjcydm = 0, where x, y are distances from the
two planes, jxydm is the mixed moment in regard to the
two planes, which are called conjugate when it vanishes.

The quotient of the second moment of a body in
regard to any plane by the mass of the body, or jx*dm : m,
is the square of a length k which is called the swing-

32 DYNAMIC.

radius of the body in regard to that plane. Thus
jx z dm = k*m. If the distances x are drawn perpendicular
to the plane, the quantity k is called the swing-radius
simply ; but if they are measured in any other direction,
the ratio fx z dm : m is the square of the swing-radius
measured in that direction. Thus the swing-radius mea-
sured in any direction is a line in that direction whose
component normal to the plane is the swing-radius k.

SWING-ELLIPSOID.

Take any plane through a point o, then a second plane
through o and the pole of the first, then a third through
o and the poles of the other two. Then each of these
planes is conjugate to the other two.

Let oX, oY, oZ be their lines of intersection, and oh
be a line making angles afiy with these lines respectively.
Then the distance from a point
p to a plane Q through o per-
pendicular to oh is equal to
ho, if ph is perpendicular on
oh. Now this is the projection
on oh of op, and therefore it
is the sum of the projections
of on, nm, mp, the coordinates
of p, which shall be called
x, y, z. These projections are
x cos a, y cos /3, z cos 7; and so

ho = x cos a 4- y cos /3 + z cos 7.

Thus the second moment of the body in regard to the
plane Q, through o perpendicular to oh, is

f(x cos a + y cos /3 + z cos ryfdm, = cos 2 a jx*dm

+ cos 2 /3 /2/ 2 dm + cos 2 7 fz*dm,
since jyzdm = jzxdm = fxydm = 0,

because each coordinate plane passes through the poles of
the other two. Let

SWING ELLIPSOIDS. 33

where m is the mass of the body, so that a, b, c are swing-
radii of the three planes, measured respectively parallel
to the three lines of intersection; and let k be the swing-
radius of the plane Q. Then

F = a 2 cos 2 a + 6 2 cos 2 /3 + c 2 cos 2 7.

Now let an ellipsoid be constructed with semi-conjugate
diameters a, b, c along oX, oY, oZ; then it may be shewn
that k is the perpendicular on the tangent plane of this
ellipsoid parallel to the plane Q.

For let uvw be tangent plane at a point q\ then the
perpendicular p on it from o is equal to

ou cos a = ov cos y8 = ow cos 7,

if a, /S, 7 are the angles which that perpendicular makes
with the axes. And on : oa = oa : ou = oa cos a : p ;

# a cos a
or - = -

Now ++ = i.

a 2 + b" + c 2
therefore a 2 cos 2 a + & 2 cos 2 /3 + c 2 cos 2 7 = p*.

Hence if parallel planes be drawn on either side of
every plane through o at distances equal to the swing-

c. 3

34 DYNAMIC.

radius of that plane, they will all touch an ellipsoid;
which is called the swing-ellipsoid at the point o.

If part of the mass of the body is negative, so that
the quantities fafdm, jy*dm, fz*dm may not all be positive,
the pairs of parallel planes may touch a hyperboloid; so that
if we take into account the possibility of negative mass we
must speak of the swing-quadric, and not of the swing-
ellipsoid. The results which depend upon negative mass
are however not susceptible of the same interpretations in
solid bodies as in the case of areas.

DETERMINATION OF THE POLE OF ANY PLANE.

The squared swing-radius of any plane is equal to that
of the parallel plane through the mass-centre together with
the square of the distance between them; all being mea-
sured in any given direction.
The proof is the same as in
two dimensions. Let h be the
distance pp between the par-
allel planes pq and p'q, and
let x be the distance of any
point o from the plane p'q'
which goes through the mass-
centre. Then the second moment of

pq =/(# + hfdm =fa;' 2 dm + 2hfxdm + tfjdm = fx*dm + h*m,

because, p'q' passing through the mass-centre, its first
moment is zero, or fxdm = 0. Hence if k be the swing-
radius of pq and a of p'q, we have F = a 2 + h*.

The pole t of the plane pq is on the line me through
the mass-centre conjugate to pq, in such a position that
me . ct = a 2 , the squared swing-radius of p'q measured
parallel to me. For we know that mt multiplied by the
first moment of pq is equal to the second moment. There-
fore

a 2 + A 2 , a?

mt = -. = h + -j- = me + ct, or me . ct = a .
h n

RELATION OF POLE TO SWING-QUADRIC. 35

The theory of poles and polar planes in regard to
quadric surfaces is precisely analogous to that of poles and
polar lines in regard to conies. From a point external to
a sphere we can draw an infinite number of tangent lines
to the sphere, which altogether form a cone touching the
sphere and having its vertex at the point. The points of
contact of all these tangent lines are in one plane, called
the polar plane of the point, which is perpendicular to
the line joining its pole to the centre, and such that the
product of the distances from the centre of the pole and
its polar plane is equal to the square of the radius. The
truth of these statements can be seen by supposing the
figure on p. 25 to be spun round the line ct, when the
circle will trace out a sphere, the lines tp, tq the tangent
cone, and pq the polar plane of t. From the same figure
it may be seen that if one point lies on the polar plane of
a second, the second point will lie on the polar plane of the
first. Hence if a series of planes be drawn through any
point inside or outside of a sphere, and cones be drawn
touching the sphere along the circles where these planes
meet it, the vertices of all these cones will lie in a plane,
called the polar plane of the point.

The last theorem may now be extended to the ellipsoid
by a homogeneous strain, and to other quadric surfaces
by first proving it for surfaces of revolution, as with the
theorems on p. 25.

This being so, we see at once that if we draw the
swing-ellipsoid at the mass-centre, the
pole of any plane in regard to the body
will be the opposite point of its pole in
regard to this ellipsoid. For if a be
the point of contact of a tangent plane
parallel to pq, ca will be the swing-
radius measured in that direction of a T
plane through c parallel to pq ; and
therefore the pole t of pq in regard to the body is so

32

36 DYNAMIC.

situated that me . ct' = ca 2 . Now the pole t' in regard to
the ellipsoid is so situated that cm . ct' = ca?. Hence t, t'
are opposite points in regard to c.

The point t may also be taken to be the pole of pq in
regard to an invisible quadric surface whose semi-diameter
in the direction cm is caJ\; because cm.ct = ca*.
This invisible surface is called the null-quadric, because
the second moment of its tangent planes is zero. In fact,
the second moment of a plane parallel to pq at a distance
h from c measured parallel to cm is m (ca 2 + h?~). Hence
if h* = ca 2 , or h = ca J 1, this is zero.

Just as a plane area may be replaced by three par-
ticles, so far as second moments are concerned, so a solid
body may be replaced by a system of four particles.
These must be so placed that each is the pole of the plane
joining the other three, and their masses must be equiva-
lent to the mass of the body, i.e., they must have the same
mass-centre as the body, and the sum of their masses
must be m. Four points such that each is the pole of the
plane joining the other three in regard to a quadric
surface, are said to form a self -conjugate tetrahedron of
that surface. The theorem is therefore that any body is
equivalent, so far as second moments are concerned, to
four particles, of the same resultant mass as the body,
placed at the vertices of a self-conjugate tetrahedron of

Let x, y, z, w be the four points ; then, in regard both
to the body and to the system of four particles, the swing-
quadric at x has the lines xy, xz, xw for a set of semi-con-
jugate diameters. For clearly the line xy from x to the
pole of xzw is the swing-radius of that plane measured in
the direction xy. Hence the two systems have the same
swing-conic at x. But the squared swing-radius of any
plane through the mass-centre is got from the squared
swing-radius of a parallel plane through x by subtracting
the squared distance between them. Hence the two

SECOND MOMENTS IN REGARD TO AN AXIS. 37

systems have the same swing-conic at their common mass-
centre; so that, since they have the same mass, their
second moments in regard to any plane are equal.

SECOND MOMENTS IN REGARD TO AN AXIS.

If two planes at right angles to one another be drawn
through any straight line, the sum
of their second moments in re-
gard to any body is called the
second moment of the body about
that line as axis.

Let oZ be the line, oZX, oZY
the two perpendicular planes, oX
and oY lines in them at right
angles to oZ ; and let a, y, z be
coordinates of a point p in regard
to this system of coordinate planes. If the body is divided
into particles, one of which having the mass 8m contains
the point p, then the second moments of the body in
regard to the two planes are fxfdm and jy*dm. Thus the
sum of them is /(a; 2 + y*) dm. If r is the perpendicular
from p on the axis oZ, we know that r 2 = of + y* ;
thus the second moment of the body about oZ is j'r^dm.
Hence, to find the second moment of a body in regard to
an axis, divide the body into particles, and multiply the
mass of each particle by the square of its distance from
the axis ; the sum of all these products will approximate
as nearly as we like to the second moment if we take the
particles small enough. It follows that the second mo-
ment about an axis is independent of the pair of perpen-
dicular planes that we take.

The quotient of this second moment by the mass of
the body is the square of a length which is called the

We may shew in the same way that the sum of the
second moments of three perpendicular planes through a
fixed point depends only on the point, and not on the
aspects of the three planes. For it is /(# 2 + y* + z*) dm ;

38 DYNAMIC.

now a? + ?/ 2 + z* is the squared distance of the point p
from the origin o, so that this quantity may be reckoned
by multiplying the mass of each particle by its squared
distance from the origin. It may be called the second
moment of the body about the point o.

It is immediately obvious that the sum of the second
moments of a body in regard to an axis and a plane which
cut at right angles is equal to the second moment of the
body about their point of intersection.

that about a parallel axis through the mass-centre together
with the squared distance between them. For let A, B be
two perpendicular planes through the axis, of which A
passes through the mass-centre ; and let B' be a plane
through the mass-centre parallel to B. Then the difference
of the second moments of the axes AB and AB' will be
the difference of the second moments of the planes B, B',
which we know to be the mass of the body multiplied by
the squared distance between them.

In a similar manner it may be shewn that the second
moment of a body about any point is equal to that about
the mass-centre together with the product of the mass
of the body by the squared distance of the point from
the mass-centre.

ELLIPSOID OF GYRATION.

Let oX, oY, oZ be the directions, and a, b, c the mag-
nitudes, of the semi-axes of the swing-ellipsoid at any
point o of the body ; then the squared swing-radius of a
plane whose normal makes angles a, /3, 7 with these axes
is a 2 cos 2 a + 6 2 cos 2 y3 + c 2 cos 2 7. We shall get the second
moment of the body about an axis through o normal to
this plane if we subtract the second moment of the plane
from the second moment of the point o. This latter
quantity is m (a 2 + b*+ c 2 ), which if we remember that

cos 2 a + cos 2 /3 + cos 2 7 = 1,
may be written

m (a 2 + b 2 + c 2 ) (cos 2 a + cos 2 /3 + cos 2 7).

CONFOCAL SURFACES. 39

Hence the squared swing-radius of the axis which makes
angles afty with oX, o Y, oZ is

tf = (a 2 + 6 2 + c 2 ) (cos 2 a + cos 2 ft + cos 2 7)

- (a 2 cos 2 a + V cos 2 ft + c 2 cos 2 7)
= (6 2 + c 2 ) cos 2 a + (c 2 + a 2 ) cos 2 ft + (a 2 + 6") cos 2 7.

Here the quantities 6 2 + c 2 , c 2 + a 2 , a 2 + & 2 are squared
them by the letters A, B, C, so that

tf = A cos 2 a + B cos 2 /3 + C cos 2 7.

From this formula it appears that if we construct an
ellipsoid having A, B, C for its squared semi-axes along
will be equal to the length of it cut off by the perpen-
dicular tangent plane to this ellipsoid. The ellipsoid so
constructed is called the ellipsoid of gyration at the
point o.

CONFOCAL SURFACES.

We have seen that when different conies have a
common centre and axes in the same direction, they will
also have the same foci when the difference of the squares
of their axes is the same. So that a conic whose squared
semi-axes are a 2 , i 2 will be confocal to a conic whose
squared semi-axes are a 2 + \, b* + X, where X is any
arbitrary quantity.

By analogy, two quadric surfaces are said to be con-
focal when they have the same centre and their axes on
the same straight lines, and when moreover the differences
of the squares of their axes are the same. Thus if one
surface has squared semi-axes a 2 , 6 2 , c 2 , and another

a 2 +X, 6 2 + X, c 2 +X,

these surfaces are said to be confocal. The conies in
which they are cut by any of the principal planes are in
fact coufocal in the sense already considered.

40 DYNAMIC.

If we begin with an ellipsoid of semi-axes a, b, c, and
gradually increase these so as to keep the differences of
their squares constant, we shall obtain a series of con-
tinually increasing ellipsoids, each entirely outside the
preceding ones. These will approach to a spherical form
as they get larger, because the ratios

approach to unity when X is made very large. The series
tends therefore as a limit to an infinitely large sphere.

If we diminish the axes of the ellipsoid, still keeping
the differences of their squares constant, we shall obtain a
series of decreasing ellipsoids, each entirely contained in
the preceding ones. These will get flatter and flatter as X
approaches the value c 2 (if c is the least of the semi-
axes) ; and then the surface takes the form of a flat
ellipse in the plane of a, b, whose squared semi-axes are
a 2 c 2 , 6 2 c 2 . This is called the focal ellipse of the whole
series of surfaces.

We may now go on to give to X a negative value
greater than c 2 , so that the squared axis c 2 + X becomes
negative. This indicates that the surface has become a
hyperboloid of one sheet, which is met in visible points by
two of its axes, but not by the third. This surface may
be regarded as starting from a flat plate, consisting of that
portion of the plane a, b which is outside of the focal
ellipse. It cuts the plane ab in an ellipse confocal with
that one,' and inside it ; because its axes go on continually
diminishing. This ellipse at last shrinks into the line
joining its two foci ; this is when b z + X vanishes, by X
becoming equal to 6 2 , (if b is the mean semi-axis, and a
the greatest). The one-sheeted hyperboloid has then
become a flat plate in the plane a, c, consisting of that
portion of the plane which is between the two branches of
the hyperbola whose squared semi-axes are

a 2 -6 2 , c 2 -6 2 .

This is called the focal hyperbola of the system of
surfaces.

CONFOCAL SURFACES. 41

Giving now to X a series of values negatively greater
than b, we shall obtain a series of two-sheeted hyperboloids,
namely that which is inside each branch of the focal
hyperbola. They cut that plane in hyperbolae confocal
to it, and lying between its branches. The two sheets of
these hyperboloids approach one another and the plane
b, c, as X approaches the value a 2 . Then they unite
into a flat plate consisting of the whole of that plane, in
which the focal conic is invisible, having the squared semi-
axes i 2 a 2 , c 2 a 2 .

After this, if we continue to increase the negative
value of X, the surface becomes wholly invisible, all three
squared semi-axes being negative ; but we must regard it
as continually increasing in size until, for an infinite value
of X, it coincides with the infinite sphere before men-
tioned.

This discussion will have made it easy to see that
each of the three series, the ellipsoids, the one-sheeted
hyperboloids, and the two-sheeted hyperboloids, sweeps
over the whole of space ; so that through every point it is
possible to draw one surface of each kind belonging to the
system.

The relation to one another of the focal ellipse and
focal hyperbola may be understood from this figure. They
are in perpendicular planes, and each passes through

the foci of the other. It is found that the sum or dif-
ference of the distances from any two fixed points on one
of them to a variable point on the other is constant.

42 DYNAMIC.

In the particular case in which two of the quantities
a, b, c are equal, all the surfaces of the system are surfaces
of revolution ; and the system is obtained by rotating the
figure of p. 30 about sh if 6 = c, or the two lesser axes
are equal, and about s^ if a = b, or the two greater axes
are equal. In the former case there are two separate foci
on the axis of revolution, and an invisible focal circle in
the equatoreal plane. In the latter case there is a real
focal circle traced out by s lt h lt and two invisible foci on
the axis of revolution.

PEINCIPAL AXES.

We shall now prove that the axes of the swing-ellipsoid
at any point (principal axes at the point) are normals to
the three surfaces confocal to the null-quadric which can be
drawn through the point.

Let a, b, c be semi-axes of the swing-ellipsoid at the
mass-centre o, and let oX, oY, oZ be their directions. Then
if k be the swing-radius of a plane at the distance p from
o, the normal to which makes angles a, /3, 7 with oX, oY,
oZ, we shall have a 2 cos 2 a + 6 2 cos 2 /3 + c 2 cos 2 7 + p 2 = k*,
which may be written & 2 (cos 2 a + cos 2 /3 + cos 2 7) because
cos 2 a + cos 2 /3 + cos 2 7 = 1 . Hence

f = (F - a 2 ) cos 2 a + (F - 6 2 ) cos 2 /3 + (tf - c 2 ) cos 2 7.

From this it follows that every plane whose swing-
semi-axes are F a 2 , F 6 2 , & 2 c 2 . This surface is con-
focal to the null-quadric, whose squared semi-axes are
a 2 , 6 2 , c 2 , and to the ellipsoid of gyration, which is
got by putting k 2 = a? + V + c 2 . Each plane touches one
surface only of the system, because no two surfaces can
have the same value of k.

Now consider a point q ; draw through it the ellipsoid,
E, of this confocal system, and its tangent plane, P. For
this plane k is greater than a, 6, or c, because k* a?,
k 2 6 2 , k z c 2 are all positive. Any increase of k will
increase the axes of the ellipsoid, and vice versa. Now if

PRINCIPAL AXES. 43