Article 38
Miscellaneous Applications.

The length of the arc of the logarithmic curve y = ax is s = M(coshu + logtanh 1 2u), in which M = 1 loga, sinhu = y M.
The length of arc of the spiral of Archimedes r = aθ is s = 1 4a(sinh2u + 2u), where sinhu = θ.
In the hyperbola x2 a2 y2 b2 = 1 the radius of curvature is
ρ = (a2 sinh2u + b2 cosh2u)3 2 ab ; in which u is the measure of the sector AOP, i.e. coshu = x a, sinhu = y b.
In an oblate spheroid, the superficial area of the zone between the equator and a parallel plane at a distance y is S = πb2(sinh2u + 2u) 2e , wherein b is the axial radius, e eccentricity, sinhu = ey p , and p parameter of generating ellipse.
The length of the arc of the parabola y2 = 2px, measured from the vertex of the curve, is l = 1 4p(sinh2u + 2u), in which sinhu = y p = tanφ, where φ is the inclination of the terminal tangent to the initial one.
The centre of gravity of this arc is given by
3lx̄ = p2(cosh3u 1),64lȳ = p2(sinh4u 4u);

and the surface of a paraboloid of revolution is S = 2πȳl.

The moment of inertia of the same arc about its terminal ordinate is I = μ xl(x 2x̄) + 1 64p3N, where μ is the mass of unit length, and
N = u 1 4sinh2u 1 4sinh4u + 1 12sinh6u.
The centre of gravity of the arc of a catenary measured from the lowest point is given by
4lȳ = c2(sinh2u + 2u),lx̄ = c2(usinhu coshu + 1),

in which u = x c ; and the moment of inertia of this arc about its terminal abscissa is

I = μc3 1 12sinh3u + 3 4sinhu ucoshu.
Applications to the vibrations of bars are given in Rayleigh, Theory of Sound, Vol. I, art. 170; to the torsion of prisms in Love, Elasticity, pp. 166–74; to the flow of heat and electricity in Byerly, Fourier Series, pp. 75–81; to wave motion in fluids in Rayleigh, Vol. I, Appendix, p. 477, and in Bassett, Hydrodynamics, arts. 120, 384; to the theory of potential in Byerly p. 135, and in Maxwell, Electricity, arts. 172–4; to Non-Euclidian geometry and many other subjects in Günther, Hyperbelfunktionen, Chaps. V and VI. Several numerical examples are worked out in Laisant, Essai sur les fonctions hyperboliques.