Article 38
Miscellaneous Applications.

1.

The length of the arc of the logarithmic curve $y=a^x$ is , in which $M=\frac{1}{loga}$, $sinhu=\frac{y}{M}$.

2.

The length of arc of the spiral of Archimedes $r=a\theta$ is , where $sinhu=\theta$.

3.

In the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ the radius of curvature is
; in which $u$ is the measure of the sector $AOP$, i.e. $coshu=\frac{x}{a}$, $sinhu=\frac{y}{b}$.

4.

In an oblate spheroid, the superficial area of the zone between the equator and a parallel plane at a distance $y$ is , wherein $b$ is the axial radius, $e$ eccentricity, $sinhu=\frac{ey}{p}$, and $p$ parameter of generating ellipse.

5.

The length of the arc of the parabola $y^2=2px$, measured from the vertex of the curve, is , in which $sinhu=\frac{y}{p}=tan\phi$, where $\phi$ is the inclination of the terminal tangent to the initial one.

6.

The centre of gravity of this arc is given by

and the surface of a paraboloid of revolution is $S=2\pi \bar{y}l$.

7.

The moment of inertia of the same arc about its terminal ordinate is , where $\mu$ is the mass of unit length, and

$$N=u-\frac{1}{4}sinh2u-\frac{1}{4}sinh4u+\frac{1}{12}sinh6u.$$

8.

The centre of gravity of the arc of a catenary measured from the lowest point is given by

in which $u=\frac{x}{c}$; and the moment of inertia of this arc about its terminal abscissa is

$$I=\mu c^3\left(\frac{1}{12}sinh3u+\frac{3}{4}sinhu-ucoshu\right).$$

9.

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.