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## Article 38Miscellaneous 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 $S=2\pi \stackrel{̄}{y}l$.

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.

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