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

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pole of the opposite side;

such a triangle is called self-

EQUIVALENT TRIAD OF PARTICLES. 29

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.

Then the squared swing-radius

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

RELATION OF POLE TO SWING-QUADRIC.

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.

EQUIVALENT TETRAD OF PARTICLES.

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

the null-quadric.

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

swing-radius about the given axis.

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.

The squared swing-radius about any axis is equal to

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

swing-radii about the axes oX, o Y, oZ. We shall denote

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

oX, oY } oZ, the swing-radius about any axis through o

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,

which start with the remaining portion of the plane a, c,

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-

radius is k touches the quadric surface whose squared

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

such a triangle is called self-

EQUIVALENT TRIAD OF PARTICLES. 29

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.

Then the squared swing-radius

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

RELATION OF POLE TO SWING-QUADRIC.

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.

EQUIVALENT TETRAD OF PARTICLES.

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

the null-quadric.

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

swing-radius about the given axis.

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.

The squared swing-radius about any axis is equal to

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

swing-radii about the axes oX, o Y, oZ. We shall denote

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

oX, oY } oZ, the swing-radius about any axis through o

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,

which start with the remaining portion of the plane a, c,

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-

radius is k touches the quadric surface whose squared

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