(1)
The tension is constant, and the pressure per unit length varies as the curvature.
Next suppose that the curve is “rough”; and let Fδs be the tangential force of friction on δs. We have δT ± Fδs = 0, Tδψ = Rδs, 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 = μR, everywhere, where μ is the coefficient of friction. If the string be on the point of slipping in the direction in which ψ increases, the lower sign is to be taken; hence δT = Fδs = μTδψ, whence
T = T0eμψ,
(2)
if T0be the tension corresponding to ψ = 0. This illustrates the resistance to dragging of a rope coiled round a post;e.g.if we put μ = .3, ψ = 2π, 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 ψ denote the inclination to the horizontal, and wδs the weight of an element δs. The tangential and normal components of wδs are −s sinψ and −wδs cosψ. Hence
δT = wδs sin ψ, Tδψ = wδs cos ψ + Rδs.
(3)
If we take rectangular axes Ox, Oy, of which Oy is drawn vertically upwards, we have δy = sin ψ δs, whence δT = wδy. 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
(5)
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 ψ = T0, T sinψ = W,
(6)
where T0is 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 = T0tan ψ. 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 ψ.
(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 ψ + tan ψ), y = a sec ψ.
(8)
Eliminating ψ we obtain the Cartesian equation
(9)
of thecommon 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 (ψ = 0) of the curve. The horizontal line through O is called thedirectrix. The relations
(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 theparameterof 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 λ of the wire. At the ends we shall have y = λ, orλ = a coshx,a(11)and the problem is to make x a maximum for variations of a. Differentiating (11) we find that, if dx/da = 0,xtanhx= 1.aa(12)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 therefore2x = 2au = 2λ/sinh u = 1.326 λ,and the length of wire is2s = 2λ/u = 1.667 λ.The tangents at the ends meet on the directrix, and their inclination to the horizontal is 56° 30′.Fig.57.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 ψ the inclination to the horizontal at A or B, we have 2Tψ = W, AB = 2ρψ, approximately, where ρ is the radius of curvature. Since 2kρ = (1⁄2AB)2, ultimately, we havek =1⁄8W · 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.
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 λ of the wire. At the ends we shall have y = λ, or
(11)
and the problem is to make x a maximum for variations of a. Differentiating (11) we find that, if dx/da = 0,
(12)
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 = 2λ/sinh u = 1.326 λ,
and the length of wire is
2s = 2λ/u = 1.667 λ.
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 ψ the inclination to the horizontal at A or B, we have 2Tψ = W, AB = 2ρψ, approximately, where ρ is the radius of curvature. Since 2kρ = (1⁄2AB)2, ultimately, we have
k =1⁄8W · 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 thecatenary 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 = wλ, where λ is a constant length. Resolving along the normal the forces on an element δs, we find Tδψ = wδs cos ψ, whenceρ =ds= λ sec ψ.dψ(15)From this we derivex = λψ, y = λ log secx,λ(16)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 = ±1⁄2πλ; this shows that however the thickness of a cable be adjusted there is a limit πλ to the horizontal span, where λ depends on the tensile strength of the material. For a uniform catenary the limit was found above to be 1.326λ.
Finally, we may refer to thecatenary 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 = wλ, where λ is a constant length. Resolving along the normal the forces on an element δs, we find Tδψ = wδs cos ψ, whence
(15)
From this we derive
(16)
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 = ±1⁄2πλ; this shows that however the thickness of a cable be adjusted there is a limit πλ to the horizontal span, where λ depends on the tensile strength of the material. For a uniform catenary the limit was found above to be 1.326λ.
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 δs 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..., Pn, with which are associated certain coefficients m1, m2, ... mn, 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 Σ(m) = 0. This has a certain interest in magnetism.
In a given mass-system there exists one and only one point G such that
Σ(m·GP>) = 0.
(1)
For, take any point O, and construct the vector
(2)
Then
Σ(m·GP>) = Σ {m(GO>+OP>)} = Σ(m)·GO>+ Σ(m)·OP>= 0.
(3)
Also there cannot be a distinct point G′ such that Σ(m·G′P) = 0, for we should have, by subtraction,
Σ {m(GP>+PG>′)} = 0, or Σ(m)·GG′ = 0;
(4)
i.e.G′ must coincide with G. The point G determined by (1) is called themass-centreorcentre of inertiaof 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, ... Pnwe draw any system of parallel planes meeting a straight line OX in the points M1, M2... Mn, the collinear vectorsOM>1,OM>2...OM>nmay be called the “projections” ofOP>1,OP>2, ...OP>non OX. Let these projections be denoted algebraically by x1, x2, ... xn, the sign being positive or negative according as the direction is that of OX or the reverse. Since the projection of a vector-sumis the sum of the projections of the several vectors, the equation (2) gives
(5)
ifxbe the projection ofOG>. Hence if the Cartesian co-ordinates of P1, P2, ... Pnrelative to any axes, rectangular or oblique be (x1, y1, z1), (x2, y2, z2), ..., (xn, yn, zn), the mass-centre (x,y,z) is determined by the formulae
(6)
If we write x =x+ ξ, y =y+ η, z =z+ ζ, so that ξ, η, ζ denote co-ordinates relative to the mass-centre G, we have from (6)
Σ(mξ) = 0, Σ(mη) = 0, Σ(mζ) = 0.
(7)
One or two special cases may be noticed. If three masses α, β, γ be situate at the vertices of a triangle ABC, the mass-centre of β and γ is at a point A′ in BC, such that β·BA′ = γ·A′C. The mass-centre (G) of α, β, γ will then divide AA′ so that α·AG = (β + γ) GA′. It is easily proved thatα : β : γ = ΔBGA : ΔGCA : ΔGAB;also, by giving suitable values (positive or negative) to the ratios α : β : γ 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 α + β + γ = 0, G is at infinity; if α = β = γ, G is at the intersection of the median lines of the triangle; if α : β : γ = a : b : c, G is at the centre of the inscribed circle. Again, if G be the mass-centre of four particles α, β, γ, δ situate at the vertices of a tetrahedron ABCD, we findα : β : γ : δ = tetnGBCD : tetnGCDA : tetnGDAB : tetnGABC,and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space. If α + β + γ + δ = O, G is at infinity; if α = β = γ = δ, G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if α : β : γ : δ = ΔBCD : ΔCDA : ΔDAB : ΔABC, 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” (seeElasticity) the new position of the mass-centre will be at that point of the strained figure which corresponds to the original mass-centre.
One or two special cases may be noticed. If three masses α, β, γ be situate at the vertices of a triangle ABC, the mass-centre of β and γ is at a point A′ in BC, such that β·BA′ = γ·A′C. The mass-centre (G) of α, β, γ will then divide AA′ so that α·AG = (β + γ) GA′. It is easily proved that
α : β : γ = ΔBGA : ΔGCA : ΔGAB;
also, by giving suitable values (positive or negative) to the ratios α : β : γ 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 α + β + γ = 0, G is at infinity; if α = β = γ, G is at the intersection of the median lines of the triangle; if α : β : γ = a : b : c, G is at the centre of the inscribed circle. Again, if G be the mass-centre of four particles α, β, γ, δ situate at the vertices of a tetrahedron ABCD, we find
α : β : γ : δ = tetnGBCD : tetnGCDA : tetnGDAB : tetnGABC,
and by suitable determination of the ratios on the left hand we can make G assume any assigned position in space. If α + β + γ + δ = O, G is at infinity; if α = β = γ = δ, G bisects the lines joining the middle points of opposite edges of the tetrahedron ABCD; if α : β : γ : δ = ΔBCD : ΔCDA : ΔDAB : ΔABC, 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” (seeElasticity) 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·OP>1, m2·OP>2, ... mn·OP>nwill have a resultant represented hy Σ(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,... mn, acting at P1, P2... Pnhave a resultant proportional to Σ(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, ... Pn, and P1′, P2′, ... Pn′ represent two configurations of the series of particles, then
Σ(m·PP>′) = Sigma(m)·GG>′,
(8)
where G, G′ are the two positions of the mass-centre. The forces m1·P>1P1′, m2·P>2P2′, ... mn·P>nPn′, considered as localized vectors, do not, however, as a rule reduce to a single resultant.
We proceed to the theory of theplane,axialandpolar quadratic momentsof the system. The axial moments have alone a dynamical significance, but the others are useful as subsidiary conceptions. If h1, h2, ... hnbe the perpendicular distances of the particles from any fixed plane, the sum Σ(mh2) is the quadratic moment with respect to the plane. If p1, p2, ... pnbe the perpendicular distances from any given axis, the sum Σ(mp2) is the quadratic moment with respect to the axis; it is also called themoment of inertiaabout the axis. If r1, r2, ... rnbe the distances from a fixed point, the sum Σ(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 Σ(m), the result is called themean squareof 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 theradius of gyrationof 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
Ix= Σ(mx2), Iy= Σ(my2), Iz= Σ(mz2);
(9)
those with respect to the co-ordinate axes are
Iyz= Σ {m (y2+ z2)}, Izx= Σ {m (z2+ x2)}, Ixy= Σ {m (x2+ y2)};
(10)
whilst the polar quadratic moment with respect to O is
I0= Σ {m (x2+ y2+ z2)}.
(11)
We note that
Iyz= Iy+ Iz, Izx= Iz+ Ix, Ixy= Ix+ Iy,
(12)
and
I0= Ix+ Iy+ Iz=1⁄2(Iyz+ Izx+ Ixy).
(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=1⁄2Ma2, where M is the mass and a the radius. Since Ix= Iy, Iz= 0, we deduce Izx=1⁄2Ma2, Ixy=1⁄2Ma2; hence the value of the squared radius of gyration is for a diameter1⁄4a2, and for the axis of symmetry1⁄2a2. Again, for a uniform solid sphere having its centre at the origin we find I0=3⁄5Ma2, Ix= Iy= Iz=1⁄5Ma2, Iyz= Izx= lxy=3⁄5Ma2;i.e.the square of the radius of gyration with respect to a diameter is2⁄5a2. 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 Ix=1⁄5Ma2, Iy=1⁄5Mb2, Iz=1⁄5Mc2, whence Iyz=1⁄5M (b2+ c2), &c.
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=1⁄2Ma2, where M is the mass and a the radius. Since Ix= Iy, Iz= 0, we deduce Izx=1⁄2Ma2, Ixy=1⁄2Ma2; hence the value of the squared radius of gyration is for a diameter1⁄4a2, and for the axis of symmetry1⁄2a2. Again, for a uniform solid sphere having its centre at the origin we find I0=3⁄5Ma2, Ix= Iy= Iz=1⁄5Ma2, Iyz= Izx= lxy=3⁄5Ma2;i.e.the square of the radius of gyration with respect to a diameter is2⁄5a2. 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 Ix=1⁄5Ma2, Iy=1⁄5Mb2, Iz=1⁄5Mc2, whence Iyz=1⁄5M (b2+ c2), &c.
If φ(x, y, z) be any homogeneous quadratic function of x, y, z, we have
Σ {mφ (x, y, z)} = Σ {mφ (x+ ξ,y+ η,z+ ζ) }= Σ {mφ (x, y, z)} + Σ {mφ (ξ, η, ζ)},
(14)
since the terms which are bilinear in respect tox,y,z, and ξ, η, ζ vanish, in virtue of the relations (7). Thus
Ix= Iξ + Σ(m)x2,
(15)
Iyz= Iηζ + Σ(m) · (y2+z2),
(16)
with similar relations, and
IO= IG+ Σ(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
(18)
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, ... Pnin succession, we obtain
(19)
If we multiply these equations by m1, m2... mn, respectively, and add, we find
ΣΣ (mrms· PrPs2) = Σ (m) · Σ (m · GP2),
(20)
provided the summation ΣΣ 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=3⁄8a2, if a be the length of an edge.
Another type of quadratic moment is supplied by thedeviation-moments, orproducts of inertiaof a distribution of matter. Thus the sum Σ(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.:—
Σ (m · yz) = Σ (m · ηζ) + Σ (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
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
(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 theprincipal axes of inertiaat 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
(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 Mk2will envelop the quadric
(31)
and the quadrics corresponding to different values of k2will be confocal. If we write
k2= a2+ b2+ c2+ θ,b2+ c2= α2, c2+ a2= β2, a2+ b2= γ2
(32)
the equation (31) becomes
(33)
for different values of θ this represents a system of quadrics confocal with the ellipsoid
(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 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 θ; and if θ1, θ2, θ3be the roots we find
θ1+ θ2+ θ3= r2− α2− β2− γ2,
(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−θ1, r2− θ2, r2− 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
(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);
(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/ 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= 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 theprincipal moments of inertiaat O. Since they are essentially positive the quadric is an ellipsoid; it is called themomental ellipsoidat 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,