up | next | prev | ptail | tail |

Miscellaneous Applications.

- 1.
- The length of the arc of the logarithmic curve $y={a}^{x}$ is $s=M(coshu+logtanh\frac{1}{2}u)$, in which $M=\frac{1}{loga}$, $sinhu=\frac{y}{M}$.
- 2.
- The length of arc of the spiral of Archimedes $r=a\theta $ is $s=\frac{1}{4}a(sinh2u+2u)$, where $sinhu=\theta $.
- 3.
- In the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$
the radius of curvature is

$\rho =\frac{{({a}^{2}{sinh}^{2}u+{b}^{2}{cosh}^{2}u)}^{\frac{3}{2}}}{ab}$; 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 $S=\frac{\pi {b}^{2}(sinh2u+2u)}{2e}$, 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 $l=\frac{1}{4}p(sinh2u+2u)$, 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
$$3l\stackrel{\u0304}{x}={p}^{2}({cosh}^{3}u-1),\phantom{\rule{1em}{0ex}}64l\stackrel{\u0304}{y}={p}^{2}(sinh4u-4u);$$ and the surface of a paraboloid of revolution is $S=2\pi \stackrel{\u0304}{y}l$.

- 7.
- The moment of inertia of the same arc about its terminal ordinate is
$I=\mu \left[xl(x-2\stackrel{\u0304}{x})+\frac{1}{64}{p}^{3}N\right]$, 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
$$4l\stackrel{\u0304}{y}={c}^{2}(sinh2u+2u),\phantom{\rule{1em}{0ex}}l\stackrel{\u0304}{x}={c}^{2}(usinhu-coshu+1),$$ 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.

up | next | prev | ptail | top |