Page:EB1911 - Volume 17.djvu/992

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

λx + μy + νz = 0,

(22)

where λ, μ, ν are direction-cosines, is

Σ {m (λx + μy + νz)²} = Σ (mx²)·λ² + Σ (my²)·μ² + Σ (mz²)·ν² + 2Σ (myz)·μν + 2Σ (mzx)·νλ + 2Σ (mxy)·λμ,

(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

Σ(myz) = 0,   Σ(mzx) = 0,   Σ(mxy) = 0;

(24)

and if we write

Σ(mx²) = Ma²,   Σ(my²) = Mb²,   Σ(mz²) = Mc²,

(25)

where M = Σ(m), the quadratic moment becomes M(a²λ² + b²μ² + c²ν²), or Mp², where p is the distance of the origin from that tangent plane of the ellipsoid

x2	+	y2	+	z2	= 1,
a2		b2		c2

(26)

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,
a′2		b′2		c′2

(27)

provided

Σ(mx′²) = Ma′²,   Σ(my′²) Mb′²,   Σ(mz′²) = Mc′²;

also that

Σ(my′z′) = 0,   Σ(mz′x′) = 0,   Σ(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 λx + μy + νz = 0 will be M(a²λ² + b²μ² + c²ν²), and that with respect to a parallel plane

λx + μy + νz = p

(29)

will be M (a²λ² + b²μ² + c²ν² + p²), by (15). This will have a given value Mk², provided

p² = (k² − a²) λ² + (k² − b²) μ² + (k² − c²) ν².

(30)

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

x2	+	y2	+	z2	= 1,
k2 − a2		k2 − b2		k2 − c2

(31)

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

k² = a² + b² + c² + θ,
b² + c² = α²,   c² + a² = β²,   a² + b² = γ²

(32)

the equation (31) becomes

x2	+	y2	+	z2	= 1;
α2 + θ		β2 + θ		γ2 + θ

(33)

for different values of θ this represents a system of quadrics confocal with the ellipsoid

x2	+	y2	+	z2	= 1,
α2		β2		γ2

(34)

which we shall meet with presently as the “ellipsoid of gyration” at G. Now consider the tangent plane ω at any point P of a confocal, the tangent plane ω′ at an adjacent point N′, and a plane ω″ through P parallel to ω′. The distance between the planes ω′ and ω″ will be of the second order of small quantities, and the quadratic moments with respect to ω′ and ω″ will therefore be equal, to the first order. Since the quadratic moments with respect to ω and ω′ are equal, it follows that ω 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 are 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 θ; and if θ₁, θ₂, θ₃ be the roots we find

θ₁ + θ₂ + θ₃ = r ² − α² − β² − γ²,

(35)

where r ² = 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 r ² −θ₁, r ² − θ₂, r ² − θ₃, 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
λ		μ		ν

(36)

is x² + y² + z² − (λx + μy + νz)², the moment of inertia about this axis is

I = Σ [m { (λ² + μ² + ν²) (x² + y² + z²) − (λx + μy + νz)²} ]
= Aλ² + Bμ² + Cν² − 2Fμν − 2Gνλ − 2Hλμ,⁠

(37)

provided

A = Σ {m (y² + z²)},   B = Σ {m (z² + x²)},   C = Σ {m (x² + y²)},
F = Σ (myz),   G = Σ (mzx),   H = Σ (mxy);$\scriptstyle {\left.{\begin{matrix}\ \\\ \end{matrix}}\right\}\,}$

(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 = Mε⁴

(39)

where ε is an arbitrary linear magnitude, the intercept r which it makes on a radius drawn in the direction λ, μ, ν is found by putting x, y, z = λr, μr, νr. Hence, by comparison with (37),

I = Mε⁴ / r ².

(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 = Mε⁴.

(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/√2.

If we write A = Mα², B = Mβ², C = Mγ², the formula (37), when referred to the principal axes at O, becomes

I = M (α²λ² + β²μ² + γ²ν²) = Mp²,

(42)

if p denotes the perpendicular drawn from O in the direction (λ, μ, ν) to a tangent plane of the ellipsoid

x2	+	y2	+	z2	= 1
α2		β2		γ2

(43)

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 = ε⁴.

(44)

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

x2	+	y2	= 1,
a2		b2

(45)

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