Page:EB1911 - Volume 17.djvu/990

which are involved in the preceding formulae are also noteworthy. It is a classical problem in the calculus of variations
Fig. 56. 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 λ of the wire. At the ends we shall have y = λ, or

λ = a cosh x/a,

(11)

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

x/a tanh x/a

(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′.

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 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 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 = wλ, where λ is a constant length. Resolving along the normal the forces on an element δs, we find Tδψ = wδs cos ψ, whence

ρ =	ds	= λ sec ψ.
	dψ

(15)

From this we derive

x = λψ,   y = λ log sec	x	,
	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.

Fig. 58.

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 . . ., 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 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

OG→ =	Σ(m·OP→)	.
	Σ(m)

(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 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 OM→₁, OM→₂ . . . OM→_n may be called the “projections” of OP→₁, OP→₂, . . . 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