Chapter XV: Part I: Statics (3)

The above method lends itself naturally to the investigation of the
_critical forms_ of a frame whose general structure is given. We have
seen that the stresses produced by an equilibrating system of
extraneous forces in a frame which is just rigid, according to the
criterion of § 6, are in general uniquely determinate; in particular,
when there are no extraneous forces the bars are in general free from
stress. It may however happen that owing to some special relation
between the lengths of the bars the frame admits of an infinitesimal
deformation. The simplest case is that of a frame of three bars, when
the three joints A, B, C fall into a straight line; a small
displacement of the joint B at right angles to AC would involve
changes in the lengths of AB, BC which are only of the second order of
small quantities. Another example is shown in fig. 53. The graphical
method leads at once to the detection of such cases. Thus in the
hexagonal frame of fig. 52, if an infinitesimal deformation is
possible without removing the bar CF, the instantaneous centre of CF
(when AB is fixed) will be at the intersection of AF and BC, and since
CC´, FF´ represent the virtual velocities of the points C, F, turned
each through a right angle, C´F´ must be parallel to CF. Conversely,
if this condition be satisfied, an infinitesimal deformation is
possible. The result may be generalized into the statement that a
frame has a critical form whenever a frame of the same structure can
be designed with corresponding bars parallel, but without complete
geometric similarity. In the case of fig. 52 it may be shown that an
equivalent condition is that the six points A, B, C, D, E, F should
lie on a conic (M. W. Crofton). This is fulfilled when the opposite
sides of the hexagon are parallel, and (as a still more special case)
when the hexagon is regular.

When a frame has a critical form it may be in a state of stress
independently of the action of extraneous forces; moreover, the
stresses due to extraneous forces are indeterminate, and may be
infinite. For suppose as before that one of the bars is removed. If
there are no extraneous forces the equation of virtual work reduces to
S·[delta]s = 0, where S is the stress in the removed bar, and [delta]s
is the change in the distance between the joints which it connected.
In a critical form we have [delta]s = 0, and the equation is satisfied
by an arbitrary value of S; a consistent system of stresses in the
remaining bars can then be found by preceding rules. Again, when
extraneous forces P act on the joints, the equation is

[Sigma](P·[delta]p) + S·[delta]s = 0,

where [delta]p is the displacement of any joint in the direction of
the corresponding force P. If [Sigma](P·[delta]p) = 0, the stresses
are merely indeterminate as before; but if [Sigma] (P·[delta]p) does
not vanish, the equation cannot be satisfied by any finite value of S,
since [delta]s = 0. This means that, if the material of the frame were
absolutely unyielding, no finite stresses in the bars would enable it
to withstand the extraneous forces. With actual materials, the frame
would yield elastically, until its configuration is no longer
"critical." The stresses in the bars would then be comparatively very
great, although finite. The use of frames which approximate to a
critical form is of course to be avoided in practice.

A brief reference must suffice to the theory of three dimensional
frames. This is important from a technical point of view, since all
structures are practically three-dimensional. We may note that a frame
of n joints which is just rigid must have 3n - 6 bars; and that the
stresses produced in such a frame by a given system of extraneous
forces in equilibrium are statically determinate, subject to the
exception of "critical forms."

§ 10. _Statics of Inextensible Chains._--The theory of bodies or structures which are deformable in their smallest parts belongs properly to elasticity (q.v.). The case of inextensible strings or chains is, however, so simple that it is generally included in expositions of pure statics.

It is assumed that the form can be sufficiently represented by a plane curve, that the stress (tension) at any point P of the curve, between the two portions which meet there, is in the direction of the tangent at P, and that the forces on any linear element [delta]s must satisfy the conditions of equilibrium laid down in § 1. It follows that the forces on any finite portion will satisfy the conditions of equilibrium which apply to the case of a rigid body (§ 4).

We will suppose in the first instance that the curve is plane. It is often convenient to resolve the forces on an element PQ (= [delta]s) in the directions of the tangent and normal respectively. If T, T + [delta]T be the tensions at P, Q, and [delta][psi] be the angle between the directions of the curve at these points, the components of the tensions along the tangent at P give (T + [delta]T) cos [psi] - T, or [delta]T, ultimately; whilst for the component along the normal at P we have (T + [delta]T) sin [delta][psi], or T[delta][psi], or T[delta]s/[rho], where [rho] is the radius of curvature.

Suppose, for example, that we have a light string stretched over a smooth curve; and let R[delta]s denote the normal pressure (outwards from the centre of curvature) on [delta]s. The two resolutions give [delta]T = 0, T[delta][psi] = R[delta]s, or

T = const., R = T/[rho]. (1)

The tension is constant, and the pressure per unit length varies as the curvature.

Next suppose that the curve is "rough"; and let F[delta]s be the tangential force of friction on [delta]s. We have [delta]T ± F[delta]s = 0, T[delta][psi] = R[delta]s, where the upper or lower sign is to be taken according to the sense in which F acts. We assume that in limiting equilibrium we have F = [mu]R, everywhere, where [mu] is the coefficient of friction. If the string be on the point of slipping in the direction in which [psi] increases, the lower sign is to be taken; hence [delta]T = F[delta]s = [mu]T[delta][psi], whence

T = T0 e^([mu][psi]), (2)

if T0 be the tension corresponding to [psi] = 0. This illustrates the resistance to dragging of a rope coiled round a post; e.g. if we put [mu] = .3, [psi] = 2[pi], we find for the change of tension in one turn T/T0 = 6.5. In two turns this ratio is squared, and so on.

Again, take the case of a string under gravity, in contact with a smooth curve in a vertical plane. Let [psi] denote the inclination to the horizontal, and w [delta]s the weight of an element [delta]s. The tangential and normal components of w[delta]s are -s sin [psi] and -w [delta]s cos [psi]. Hence

[delta]T = w [delta]s sin [psi], T [delta][psi] = w [delta]s cos [psi] + R[delta]s. (3)

If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have [delta]y = sin[psi] [delta]s, whence [delta]T = w[delta]y. If the string be uniform, w is constant, and

T = wy + const. = w(y - y0), (4)

say; hence the tension varies as the height above some fixed level (y0). The pressure is then given by the formula

d[psi]
R = T ------ - w cos [psi]. (5)
ds

In the case of a chain hanging freely under gravity it is usually convenient to formulate the conditions of equilibrium of a finite portion PQ. The forces on this reduce to three, viz. the weight of PQ and the tensions at P, Q. Hence these three forces will be concurrent, and their ratios will be given by a triangle of forces. In particular, if we consider a length AP beginning at the lowest point A, then resolving horizontally and vertically we have

T cos [psi] = T0, T sin [psi] = W, (6)

where T0 is the tension at A, and W is the weight of PA. The former equation expresses that the horizontal tension is constant.

If the chain be uniform we have W = ws, where s is the arc AP: hence ws = T0 tan[psi]. If we write T0 = wa, so that a is the length of a portion of the chain whose weight would equal the horizontal tension, this becomes

s = a tan [psi]. (7)

This is the "intrinsic" equation of the curve. If the axes of x and y be taken horizontal and vertical (upwards), we derive

x = a log (sec [psi] + tan [psi]), y = a sec [psi]. (8)

Eliminating [psi] we obtain the Cartesian equation

x
y = a cosh --- (9)
a

of the _common catenary_, as it is called (fig. 56). The omission of the additive arbitrary constants of integration in (8) is equivalent to a special choice of the origin O of co-ordinates; viz. O is at a distance a vertically below the lowest point ([psi] = 0) of the curve. The horizontal line through O is called the _directrix_. The relations

s = a sinh x/a, y² = a² + s², T = T0 sec [psi] = wy, (10)

which are involved in the preceding formulae are also noteworthy. It is a classical problem in the calculus of variations to deduce the equation (9) from the condition that the depth of the centre of gravity of a chain of given length hanging between fixed points must be stationary (§ 9). The length a is called the _parameter_ of the catenary; it determines the scale of the curve, all catenaries being geometrically similar. If weights be suspended from various points of a hanging chain, the intervening portions will form arcs of equal catenaries, since the horizontal tension (wa) is the same for all. Again, if a chain pass over a perfectly smooth peg, the catenaries in which it hangs on the two sides, though usually of different parameters, will have the same directrix, since by (10) y is the same for both at the peg.

As an example of the use of the formulae we may determine the maximum
span for a wire of given material. The condition is that the tension
must not exceed the weight of a certain length [lambda] of the wire.
At the ends we shall have y = [lambda], or

x
[lambda] = a cosh ---, (11)
a

and the problem is to make x a maximum for variations of a.
Differentiating (11) we find that, if dx/da = 0,

x x
--- tanh --- = 1. (12)
a a

It is easily seen graphically, or from a table of hyperbolic tangents,
that the equation u tanh u = 1 has only one positive root (u = 1.200);
the span is therefore

2x = 2au = 2[lambda]/sinh u = 1.326[lambda],

and the length of wire is

2s = 2[lambda]/u = 1.667 [lambda].

The tangents at the ends meet on the directrix, and their inclination
to the horizontal is 56° 30´.

The relation between the sag, the tension, and the span of a wire
(e.g. a telegraph wire) stretched nearly straight between two points
A, B at the same level is determined most simply from first
principles. If T be the tension, W the total weight, k the sag in the
middle, and [psi] the inclination to the horizontal at A or B, we have
2T[psi] = W, AB = 2[rho][psi], approximately, where [rho] is the
radius of curvature. Since 2k[rho] = (½AB)², ultimately, we have

k = (1/8)W·AB/T. (13)

The same formula applies if A, B be at different levels, provided k be
the sag, measured vertically, half way between A and B.

In relation to the theory of suspension bridges the case where the weight of any portion of the chain varies as its horizontal projection is of interest. The vertical through the centre of gravity of the arc AP (see fig. 55) will then bisect its horizontal projection AN; hence if PS be the tangent at P we shall have AS = SN. This property is characteristic of a parabola whose axis is vertical. If we take A as origin and AN as axis of x, the weight of AP may be denoted by wx, where w is the weight per unit length at A. Since PNS is a triangle of forces for the portion AP of the chain, we have wx/T0 = PN/NS, or

y = w·x²/2T0, (14)

which is the equation of the parabola in question. The result might of course have been inferred from the theory of the parabolic funicular in § 2.

Finally, we may refer to the _catenary of uniform strength_, where the
cross-section of the wire (or cable) is supposed to vary as the
tension. Hence w, the weight per foot, varies as T, and we may write
T = w[lambda], where [lambda] is a constant length. Resolving along
the normal the forces on an element [delta]s, we find T[delta][psi] =
w[delta]s cos[psi], whence

ds
p = ------ = [lambda] sec [psi]. (15)
d[psi]

From this we derive

x
x = [lambda][psi], y = [lambda] log sec --------, (16)
[lambda]

where the directions of x and y are horizontal and vertical, and the
origin is taken at the lowest point. The curve (fig. 58) has two
vertical asymptotes x = ± ½[pi][lambda]; this shows that however the
thickness of a cable be adjusted there is a limit [pi][lambda] to the
horizontal span, where [lambda] depends on the tensile strength of the
material. For a uniform catenary the limit was found above to be
1.326[lambda].

For investigations relating to the equilibrium of a string in three dimensions we must refer to the textbooks. In the case of a string stretched over a smooth surface, but in other respects free from extraneous force, the tensions at the ends of a small element [delta]s must be balanced by the normal reaction of the surface. It follows that the osculating plane of the curve formed by the string must contain the normal to the surface, i.e. the curve must be a "geodesic," and that the normal pressure per unit length must vary as the principal curvature of the curve.

§ 11. _Theory of Mass-Systems._--This is a purely geometrical subject. We consider a system of points P1, P2 ..., P_n, with which are associated certain coefficients m1, m2, ... m_n, respectively. In the application to mechanics these coefficients are the masses of particles situate at the respective points, and are therefore all positive. We shall make this supposition in what follows, but it should be remarked that hardly any difference is made in the theory if some of the coefficients have a different sign from the rest, except in the special case where [Sigma](m) = 0. This has a certain interest in magnetism.

In a given mass-system there exists one and only one point G such that

[Sigma](m·[->GP]) = 0. (1)

For, take any point O, and construct the vector

[Sigma](m·[->OP])
[->OG] = -----------------. (2)
[Sigma](m)

Then

[Sigma](m·[->GP]) = [Sigma]{m([->GO] + [->OP])} = [Sigma](m)·[->GO] + [Sigma](m)·[->OP] = 0. (3)

Also there cannot be a distinct point G´ such that [Sigma](m·G´P) = 0, for we should have, by subtraction,

[Sigma]{m([->GP] + [->PG´])} = 0, or [Sigma](m)·GG´ = 0; (4)

i.e. G´ must coincide with G. The point G determined by (1) is called the _mass-centre_ or _centre of inertia_ of the given system. It is easily seen that, in the process of determining the mass-centre, any group of particles may be replaced by a single particle whose mass is equal to that of the group, situate at the mass-centre of the group.

If through P1, P2, ... P_n we draw any system of parallel planes meeting a straight line OX in the points M1, M2 ... M_n, the collinear vectors [->OM1], [->OM2] ... [->OM_n] may be called the "projections" of [->OP1], [->OP2], ... [->OP_n] on OX. Let these projections be denoted algebraically by x1, x2, ... x_n, the sign being positive or negative according as the direction is that of OX or the reverse. Since the projection of a vector-sum is the sum of the projections of the several vectors, the equation (2) gives

[Sigma](mx)
[|x] = -----------, (5)
[Sigma](m)

if [|x] be the projection of [->OG]. Hence if the Cartesian co-ordinates of P1, P2, ... P_n relative to any axes, rectangular or oblique be (x1, y1, z1), (x2, y2, z2), ..., (x_n, y_n, z_n), the mass-centre ([|x], [|y], [|z]) is determined by the formulae

[Sigma](mx) [Sigma](my) [Sigma](mz)
[|x] = -----------, [|y] = -----------, [|z] = -----------. (6)
[Sigma](m) [Sigma](m) [Sigma](m)

If we write x = [|x] + [xi], y = [|y] + [eta], z = [|z] + [zeta], so that [xi], [eta], [zeta] denote co-ordinates relative to the mass-centre G, we have from (6)

[Sigma](m[xi]) = 0, [Sigma](m[eta]) = 0, [Sigma](m[zeta]) = 0. (7)

One or two special cases may be noticed. If three masses [alpha],
[beta], [gamma] be situate at the vertices of a triangle ABC, the
mass-centre of [beta] and [gamma] is at a point A´ in BC, such that
[beta]·BA´ = [gamma]·A´C. The mass-centre (G) of [alpha], [beta],
[gamma] will then divide AA´ so that [alpha]·AG = ([beta] + [gamma])
GA´. It is easily proved that

[alpha] : [beta] : [gamma] = [Delta]BGA : [Delta]GCA : [Delta]GAB;

also, by giving suitable values (positive or negative) to the ratios
[alpha] : [beta] : [gamma] we can make G assume any assigned position
in the plane ABC. We have here the origin of the "barycentric
co-ordinates" of Möbius, now usually known as "areal" co-ordinates. If
[alpha] + [beta] + [gamma] = 0, G is at infinity; if [alpha] = [beta]
= [gamma], G is at the intersection of the median lines of the
triangle; if [alpha] : [beta] : [gamma] = a : b : c, G is at the
centre of the inscribed circle. Again, if G be the mass-centre of four
particles [alpha], [beta], [gamma], [delta] situate at the vertices of
a tetrahedron ABCD, we find

[alpha] : [beta] : [gamma] : [delta] = tet^n GBCD : tet^n GCDA : tet^n GDAB : tet^n GABC,

and by suitable determination of the ratios on the left hand we can
make G assume any assigned position in space. If [alpha] + [beta] +
[gamma] + [delta] = O, G is at infinity; if [alpha] = [beta] = [gamma]
= [delta], G bisects the lines joining the middle points of opposite
edges of the tetrahedron ABCD; if [alpha] : [beta] : [gamma] : [delta]
= [Delta]BCD : [Delta]CDA : [Delta]DAB : [Delta]ABC, G is at the
centre of the inscribed sphere.

If we have a continuous distribution of matter, instead of a system of
discrete particles, the summations in (6) are to be replaced by
integrations. Examples will be found in textbooks of the calculus and
of analytical statics. As particular cases: the mass-centre of a
uniform thin triangular plate coincides with that of three equal
particles at the corners; and that of a uniform solid tetrahedron
coincides with that of four equal particles at the vertices. Again,
the mass-centre of a uniform solid right circular cone divides the
axis in the ratio 3 : 1; that of a uniform solid hemisphere divides
the axial radius in the ratio 3 : 5.

It is easily seen from (6) that if the configuration of a system of
particles be altered by "homogeneous strain" (see ELASTICITY) the new
position of the mass-centre will be at that point of the strained
figure which corresponds to the original mass-centre.

The formula (2) shows that a system of concurrent forces represented by m1·[->OP1], m2·[->OP2], ... m_n·[->OP_n] will have a resultant represented hy [Sigma](m)·[->OG]. If we imagine O to recede to infinity in any direction we learn that a system of parallel forces proportional to m1, m2,... m_n, acting at P1, P2 ... P_n have a resultant proportional to [Sigma](m) which acts always through a point G fixed relatively to the given mass-system. This contains the theory of the "centre of gravity" (§§ 4, 9). We may note also that if P1, P2, ... P_n, and P1´, P2´, ... P_n´ represent two configurations of the series of particles, then

[Sigma](m·[->PP´]) = Sigma(m)·[->GG´], (8)

where G, G´ are the two positions of the mass-centre. The forces m1·[->P1P1´], m2·[->P2P2´], ... m_n·[->P_nP_n´], considered as localized vectors, do not, however, as a rule reduce to a single resultant.

We proceed to the theory of the _plane_, _axial_ and _polar quadratic moments_ of the system. The axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions. If h1, h2, ... h_n be the perpendicular distances of the particles from any fixed plane, the sum [Sigma](mh²) is the quadratic moment with respect to the plane. If p1, p2, ... p_n be the perpendicular distances from any given axis, the sum [Sigma](mp²) is the quadratic moment with respect to the axis; it is also called the _moment of inertia_ about the axis. If r1, r2, ... r_n be the distances from a fixed point, the sum [Sigma](mr²) is the quadratic moment with respect to that point (or pole). If we divide any of the above quadratic moments by the total mass [Sigma](m), the result is called the _mean square_ of the distances of the particles from the respective plane, axis or pole. In the case of an axial moment, the square root of the resulting mean square is called the _radius of gyration_ of the system about the axis in question. If we take rectangular axes through any point O, the quadratic moments with respect to the co-ordinate planes are

I_x = [Sigma](mx²), I_y = [Sigma](my²), I_z = [Sigma](mz²); (9)

those with respect to the co-ordinate axes are

I_yz = [Sigma]{m(y² + z²)}, I_zx = [Sigma]{m(z² + x²)},
I_xy = [Sigma]{m(x² + y²)}; (10)

whilst the polar quadratic moment with respect to O is

I0 = [Sigma]{m(x² + y² + z²)}. (11)

We note that

I_yz = I_y + I_z, I_zx = I_z + I_x, I_xy = I_x + I_y, (12)

and

I0 = I_x + I_y + I_z = ½(I_yz + I_zx + I_xy). (13)

In the case of continuous distributions of matter the summations in
(9), (10), (11) are of course to be replaced by integrations. For a
uniform thin circular plate, we find, taking the origin at its centre,
and the axis of z normal to its plane, I0 = ½Ma², where M is the mass
and a the radius. Since I_x = I_y, I_z = 0, we deduce I_zx = ½Ma²,
I_xy = ½Ma²; hence the value of the squared radius of gyration is for
a diameter ¼a², and for the axis of symmetry ½a². Again, for a uniform
solid sphere having its centre at the origin we find I0 = (3/5)Ma²,
I_x = I_y = I_z = (1/5)Ma², I_yz = I_zx = l_xy = (3/5)Ma²; i.e. the
square of the radius of gyration with respect to a diameter is
(2/5)a². The method of homogeneous strain can be applied to deduce the
corresponding results for an ellipsoid of semi-axes a, b, c. If the
co-ordinate axes coincide with the principal axes, we find I_x =
(1/5)Ma², I_y = (1/5)Mb², I_z = (1/5)Mc², whence I_yz = (1/5)M (b² +
c²), &c.

If [phi](x, y, z) be any homogeneous quadratic function of x, y, z, we have

[Sigma]{m[phi](x, y, z)} = [Sigma] {m[phi]([|x] + [xi], [|y] + [eta], [|z] + [zeta])}
= [Sigma] {m[phi](x, y, z)} + [Sigma]{m[phi]([xi], [eta], [zeta])}, (14)

since the terms which are bilinear in respect to [|x], [|y], [|z], and [xi], [eta], [zeta] vanish, in virtue of the relations (7). Thus

I_x = I[xi] + [Sigma](m)x², (15)

I_yz = I[eta][zeta] + [Sigma](m)·(y² + z²), (16)

with similar relations, and

I_O = I_G + [Sigma](m)·OG². (17)

The formula (16) expresses that the squared radius of gyration about any axis (Ox) exceeds the squared radius of gyration about a parallel axis through G by the square of the distance between the two axes. The formula (17) is due to J. L. Lagrange; it may be written

[Sigma](m·OP²) [Sigma](m·GP²)
-------------- = -------------- + OG², (18)
[Sigma](m) [Sigma](m)

and expresses that the mean square of the distances of the particles from O exceeds the mean square of the distances from G by OG². The mass-centre is accordingly that point the mean square of whose distances from the several particles is least. If in (18) we make O coincide with P1, P2, ... P_n in succession, we obtain

0 + m2·P1P2² + ... + mn·P1P_n² = [Sigma](m·GP²) + [Sigma](m)·GP1², \
m1·P2P1² + 0 + ... + mn·P2P_n² = [Sigma](m·GP²) + [Sigma](m)·GP2², > (19)
... ... ... ... ... |
m1·P_nP1² + m2·P_nP2² + ... + 0 = [Sigma](m·GP²) + [Sigma](m)·GP_n². /

If we multiply these equations by m1, m2 ... m_n, respectively, and add, we find

[Sigma][Sigma](m_r m_s·P_r P_s²) = [Sigma](m)·[Sigma](m·GP²), (20)

provided the summation [Sigma][Sigma] on the left hand be understood to include each pair of particles once only. This theorem, also due to Lagrange, enables us to express the mean square of the distances of the particles from the centre of mass in terms of the masses and mutual distances. For instance, considering four equal particles at the vertices of a regular tetrahedron, we can infer that the radius R of the circumscribing sphere is given by R² = (3/8)a², if a be the length of an edge.

Another type of quadratic moment is supplied by the _deviation-moments_, or _products of inertia_ of a distribution of matter. Thus the sum [Sigma](m·yz) is called the "product of inertia" with respect to the planes y = 0, z = 0. This may be expressed In terms of the product of inertia with respect to parallel planes through G by means of the formula (14); viz.:--

[Sigma](m·yz) = [Sigma](m·[eta][zeta]) + [Sigma](m)·yz (21)

The quadratic moments with respect to different planes through a fixed point O are related to one another as follows. The moment with respect to the plane

[lambda]x + [mu]y + [nu]z = 0, (22)

where [lambda], [mu], [nu] are direction-cosines, is

[Sigma]{(m([lambda]x + [mu]y + [nu]z)²} = [Sigma](mx²)·[lambda]² + [Sigma](my²)·[mu]² + [Sigma](mz²)·[nu]²
+ 2[Sigma](myz)·[mu][nu] + 2[Sigma](mzx)·[nu][lambda] + 2[Sigma](mxy)·[lambda][mu], (23)

and therefore varies as the square of the perpendicular drawn from O to a tangent plane of a certain quadric surface, the tangent plane in question being parallel to (22). If the co-ordinate axes coincide with the principal axes of this quadric, we shall have

[Sigma](myz) = 0, [Sigma](mzx) = 0, [Sigma](mxy) = 0; (24)

and if we write

[Sigma](mx²) = Ma², [Sigma](my²) = Mb², [Sigma](mz²) = Mc², (25)

where M = [Sigma](m), the quadratic moment becomes M(a²[lambda]² + b²[mu]² + c²[nu]²), or Mp², where p is the distance of the origin from that tangent plane of the ellipsoid

x² y² z²
--- + --- + --- = 1, (26)
a² b² c²

which is parallel to (22). It appears from (24) that through any assigned point O three rectangular axes can be drawn such that the product of inertia with respect to each pair of co-ordinate planes vanishes; these are called the _principal axes of inertia_ at O. The ellipsoid (26) was first employed by J. Binet (1811), and may be called "Binet's Ellipsoid" for the point O. Evidently the quadratic moment for a variable plane through O will have a "stationary" value when, and only when, the plane coincides with a principal plane of (26). It may further be shown that if Binet's ellipsoid be referred to any system of conjugate diameters as co-ordinate axes, its equation will be

x´² y´² z´²
--- + --- + --- = 1, (27)
a´² b´² c´²

provided

[Sigma](mx´²) = Ma´², [Sigma](my´²) Mb´², [Sigma](mz´²) = Mc´²;

also that

[Sigma](my´z´) = 0, [Sigma](mz´x´) = 0, [Sigma](mx´y´) = 0. (28)

Let us now take as co-ordinate axes the principal axes of inertia at the mass-centre G. If a, b, c be the semi-axes of the Binet's ellipsoid of G, the quadratic moment with respect to the plane [lambda]x + [mu]y + [nu]z = 0 will be M(a²[lambda]² + b²[mu]² + c²[nu]²), and that with respect to a parallel plane

[lambda]x + [mu]y + [nu]z = p (29)

will be M(a²[lambda]² + b²[mu]² + c²[nu]² + p²), by (15). This will have a given value Mk², provided

p² = (k² - a²)[lambda]² + (k² - b²)[mu]² + (k² - c²)[nu]². (30)

Hence the planes of constant quadratic moment Mk² will envelop the quadric

x² y² z²
------- + ------- + ------- = 1, (31)
k² - a² k² - b² k² - c²

and the quadrics corresponding to different values of k² will be confocal. If we write

k² = a² + b² + c² + [theta],
b² + c² = [alpha]², c² + a² = [beta]², a² + b² = [gamma]² (32)

the equation (31) becomes

x² y² z²
------------------ + ----------------- + ------------------ = 1 (33)
[alpha]² + [theta] [beta]² + [theta] [gamma]² + [theta]

for different values of [theta] this represents a system of quadrics confocal with the ellipsoid

x² y² z²
-------- + ------- + -------- = 1, (34)
[alpha]² [beta]² [gamma]²

which we shall meet with presently as the "ellipsoid of gyration" at G. Now consider the tangent plane [omega] at any point P of a confocal, the tangent plane [omega]´ at an adjacent point N´, and a plane [omega]´´ through P parallel to [omega]´. The distance between the planes [omega]´ and [omega]´´ will be of the second order of small quantities, and the quadratic moments with respect to [omega]´ and [omega]´´ will therefore be equal, to the first order. Since the quadratic moments with respect to [omega] and [omega]´ are equal, it follows that [omega] is a plane of stationary quadratic moment at P, and therefore a principal plane of inertia at P. In other words, the principal axes of inertia at P arc the normals to the three confocals of the system (33) which pass through P. Moreover if x, y, z be the co-ordinates of P, (33) is an equation to find the corresponding values of [theta]; and if [theta]1, [theta]2, [theta]3 be the roots we find

[theta]1 + [theta]2 + [theta]3 = r² - [alpha]² - [beta]² -[gamma]², (35)

where r² = x² + y² + z². The squares of the radii of gyration about the principal axes at P may be denoted by k2² + k3², k3² + k1², k1² + k2²; hence by (32) and (35) they are r² - [theta]1, r² - [theta]2, r² - [theta]3, respectively.

To find the relations between the moments of inertia about different axes through any assigned point O, we take O as origin. Since the square of the distance of a point (x, y, z) from the axis

x y z
-------- = ---- = ---- (36)
[lambda] [mu] [nu]

is x² + y² + z² - ([lambda]x + [mu]y + [nu]z)², the moment of inertia about this axis is

I = [Sigma][m{([lambda]² + [mu]² + [nu]²)(x² + y² + z²) - ([lambda]x + [mu]y + [nu]z)²}]
= A[lambda]² + B[mu]² + C[nu]² - 2F[mu][nu] - 2G[nu][lambda] - 2H[lambda][mu], (37)

provided

A = [Sigma]{m(y² + z²)}, B = [Sigma]{m(z² + x²)}, C = [Sigma]{m(x² + y²)},
F = [Sigma](myz), G = [Sigma](mzx), H = [Sigma](mxy); (38)

i.e. A, B, C are the moments of inertia about the co-ordinate axes, and F, G, H are the products of inertia with respect to the pairs of co-ordinate planes. If we construct the quadric

Ax² + By² + Cz² - 2Fyz - 2Gzx - 2Hxy = M[epsilon]^4 (39)

where [epsilon] is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction [lambda], [mu], [nu] is found by putting x, y, z = [lambda]r, [mu]r, [nu]r. Hence, by comparison with (37),

I = M[epsilon]^4/r². (40)

The moment of inertia about any radius of the quadric (39) therefore varies inversely as the square of the length of this radius. When referred to its principal axes, the equation of the quadric takes the form

Ax² + By² + Cz² = M[epsilon]^4. (41)

The directions of these axes are determined by the property (24), and therefore coincide with those of the principal axes of inertia at O, as already defined in connexion with the theory of plane quadratic moments. The new A, B, C are called the _principal moments of inertia_ at O. Since they are essentially positive the quadric is an ellipsoid; it is called the _momental ellipsoid_ at O. Since, by (12), B + C > A, &c., the sum of the two lesser principal moments must exceed the greatest principal moment. A limitation is thus imposed on the possible forms of the momental ellipsoid; e.g. in the case of symmetry about an axis it appears that the ratio of the polar to the equatorial diameter of the ellipsoid cannot be less than 1/[root]2.

If we write A = M[alpha]², B = M[beta]², C = M[gamma]², the formula (37), when referred to the principal axes at O, becomes

I = M([alpha]²[lambda]² + [beta]²[mu]² + [gamma]²[nu]²) = Mp², (42)

if p denotes the perpendicular drawn from O in the direction ([lambda], [mu], [nu]) to a tangent plane of the ellipsoid

x² y² z²
-------- + ------- + -------- = 1 (43)
[alpha]² [beta]² [gamma]²

This is called the _ellipsoid of gyration_ at O; it was introduced into the theory by J. MacCullagh. The ellipsoids (41) and (43) are reciprocal polars with respect to a sphere having O as centre.

If A = B = C, the momental ellipsoid becomes a sphere; all axes through O are then principal axes, and the moment of inertia is the same for each. The mass-system is then said to possess kinetic symmetry about O.

If all the masses lie in a plane (z = 0) we have, in the notation of
(25), c² = 0, and therefore A = Mb², B = Ma², C = M(a² + b²), so that
the equation of the momental ellipsoid takes the form

b²x² + a²y² + (a² + b²)z² = [epsilon]^4. (44)

The section of this by the plane z = 0 is similar to

x² y²
---- + ---- = 1, (45)
a² b²

which may be called the _momental ellipse_ at O. It possesses the
property that the radius of gyration about any diameter is half the
distance between the two tangents which are parallel to that diameter.
In the case of a uniform triangular plate it may be shown that the
momental ellipse at G is concentric, similar and similarly situated

to the ellipse which touches the sides of the triangle at their middle
points.

The graphical methods of determining the moment of inertia of a plane
system of particles with respect to any line in its plane may be
briefly noticed. It appears from § 5 (fig. 31) that the linear moment
of each particle about the line may be found by means of a funicular
polygon. If we replace the mass of each particle by its moment, as
thus found, we can in like manner obtain the quadratic moment of the
system with respect to the line. For if the line in question be the
axis of y, the first process gives us the values of mx, and the second
the value of [Sigma](mx·x) or [Sigma](mx²). The construction of a
second funicular may be dispensed with by the employment of a
planimeter, as follows. In fig. 59 p is the line with respect to which
moments are to be taken, and the masses of the respective particles
are indicated by the corresponding segments of a line in the
force-diagram, drawn parallel to p. The funicular ZABCD ...
corresponding to any pole O is constructed for a system of forces
acting parallel to p through the positions of the particles and
proportional to the respective masses; and its successive sides are
produced to meet p in the points H, K, L, M, ... As explained in § 5,
the moment of the first particle is represented on a certain scale by
HK, that of the second by KL, and so on. The quadratic moment of the
first particle will then be represented by twice the area AHK, that of
the second by twice the area BKL, and so on. The quadratic moment of
the whole system is therefore represented by twice the area AHEDCBA.
Since a quadratic moment is essentially positive, the various areas
are to taken positive in all cases. If k be the radius of gyration
about p we find

k² = 2 × area AHEDCBA × ON ÷ [alpha][beta],

where [alpha][beta] is the line in the force-diagram which represents
the sum of the masses, and ON is the distance of the pole O from this
line. If some of the particles lie on one side of p and some on the
other, the quadratic moment of each set may be found, and the results
added. This is illustrated in fig. 60, where the total quadratic
moment is represented by the sum of the shaded areas. It is seen that
for a given direction of p this moment is least when p passes through
the intersection X of the first and last sides of the funicular; i.e.
when p goes through the mass-centre of the given system; cf. equation
(15).