A Treatise on Electricity and Magnetism/Part I/Chapter X

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A Treatise on Electricity and Magnetism by James Clerk Maxwell Part I, Chapter X: Confocal Surfaces of the Second Degree

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CHAPTER X.

CONFOCAL QUADRIC SURFACES.

147.] Let the general equation of a confocal system be

$\frac{x^{2}}{\lambda^{2}-a^{2}}+\frac{y^{2}}{\lambda^{2}-b^{2}}+\frac{z^{2}}{\lambda^{2}-c^{2}}=1,$

(1)

where λ is a variable parameter, which we shall distinguish by the suffix λ₁ for the hyperboloids of two sheets, λ₂ for the hyperboloids of one sheet, and λ₃ for the ellipsoids. The quantities

$a,\ \lambda_{1},\ b,\ \lambda_{2},\ c,\ \lambda_{3}$

are in ascending order of magnitude. The quantity a is introduced for the sake of symmetry, but in our results we shall always suppose a = 0.

If we consider the three surfaces whose parameters are $\lambda_{1},\ \lambda_{2},\ \lambda_{3}$, we find, by elimination between their equations, that the value of x² at their point of intersection satisfies the equation

$x^{2}\left(b^{2}-a^{2}\right)\left(c^{2}-a^{2}\right)=\left(\lambda_{1}^{2}-a^{2}\right)\left(\lambda_{2}^{2}-a^{2}\right)\left(\lambda_{3}^{2}-a^{2}\right).$

(2)

The values of y² and z² may be found by transposing a, b, c symmetrically.

Differentiating this equation with respect to λ₁, we find

$\frac{dx}{d\lambda_{1}}=\frac{\lambda_{1}}{\lambda_{1}^{2}-a^{2}}x.$

(3)

If ds₁ is the length of the intercept of the curve of intersection of λ₂ and λ₃ cut off between the surfaces λ₁ and λ₁ + dλ₁ then

$\left(\frac{ds_{1}}{d\lambda_{1}}\right)^{2}=\left(\frac{dx}{d\lambda_{1}}\right)^{2}+\left(\frac{dy}{d\lambda_{1}}\right)^{2}+\left(\frac{dz}{d\lambda_{1}}\right)^{2}=\frac{\lambda_{1}^{2}\left(\lambda_{2}^{2}-\lambda_{1}^{2}\right)\left(\lambda_{3}^{2}-\lambda_{1}^{2}\right)}{\left(\lambda_{1}^{2}-a^{2}\right)\left(\lambda_{1}^{2}-b^{2}\right)\left(\lambda_{1}^{2}-c^{2}\right)}$

(4)

The denominator of this fraction is the product of the squares of the semi-axes of the surface λ₁.

If we put

$D_{1}^{2}=\lambda_{3}^{2}-\lambda_{2}^{2},\ D_{2}^{2}=\lambda_{3}^{2}-\lambda_{1}^{2},\ \mathrm{and}\ D_{3}^{2}=\lambda_{2}^{2}-\lambda_{1}^{2},$

(5)

and if we make a = 0, then

$\frac{ds_{1}}{d\lambda_{1}}=\frac{D_{2}D_{3}}{\sqrt{b^{2}-\lambda_{1}^{2}}\sqrt{c^{2}-\lambda_{1}^{2}}}$

(6)

It is easy to see that D₂ and D₃ are the semi-axes of the central section of λ₁ which is conjugate to the diameter passing through the given point, and that D₂ is parallel to ds₂, and D₃ to Ds₃.

If we also substitute for the three parameters λ₁,λ₁,λ₁ their values in terms of three functions $\alpha,\ \beta,\ \gamma$ defined by the equations

$\begin{array}{lll} \frac{da}{d\lambda_{1}}=\frac{c}{\sqrt{b^{2}-\lambda_{1}^{2}}\sqrt{c^{2}-\lambda_{1}^{2}}} & & \lambda_{1}=0\ \mathrm{when}\ \alpha=0,\\ \\\frac{d\beta}{d\lambda_{2}}=\frac{c}{\sqrt{\lambda_{2}^{2}-b^{2}}\sqrt{c^{2}-\lambda_{2}^{2}}} & & \lambda_{2}=b\ \mathrm{when}\ \beta=0,\\ \\\frac{d\gamma}{d\lambda_{3}}=\frac{c}{\sqrt{\lambda_{3}^{2}-b^{2}}\sqrt{\lambda_{3}^{2}-c^{2}}} & & \lambda_{3}=c\ \mathrm{when}\ \gamma=0;\end{array}$

(7)

then

$ds_{1}=\frac{1}{c}D_{2}D_{3}d\alpha,\ ds_{2}=\frac{1}{c}D_{3}D_{1}d\beta,\ ds_{3}=\frac{1}{c}D_{1}D_{2}d\gamma.$

(8)

148.] Now let V be the potential at any point $\alpha,\ \beta,\ \gamma$, then the resultant force in the direction of ds₁ is

$R_{1}=-\frac{dV}{ds_{1}}=-\frac{dV}{d\alpha}\frac{d\alpha}{ds_{1}}=-\frac{dV}{d\alpha}\frac{c}{D_{2}D_{3}}.$

(9)

Since $ds_{1},\ ds_{2}$, and ds₃ are at right angles to each other, the surface-integral over the element of area $ds_{2}\ ds_{3}$ is

$\begin{array}{ll} R_{1}ds_{2}ds_{3} & =\frac{dV}{d\alpha}\frac{c}{D_{2}D_{3}}\cdot\frac{D_{3}D_{1}}{c}\cdot\frac{D_{1}D_{2}}{c}\cdot d\beta\ d\gamma\\ \\ & =\frac{dV}{d\alpha}\frac{D_{1}^{2}}{c}d\beta\ d\gamma.\end{array}$

(10)

Now consider the element of volume intercepted between the surfaces $\alpha,\ \beta,\ \gamma$, and $\alpha+d\alpha,\ \beta+d\beta,\ \gamma+d\gamma.$. There will be eight such elements, one in each octant of space.

We have found the surface-integral for the element of surface intercepted from the surface α by the surfaces β and β + dβ, γ and γ + dγ.

The surface-integral for the corresponding element of the surface α + dα will be

$\frac{dV}{d\alpha}\frac{D_{1}^{2}}{c}d\beta\ d\gamma+\frac{d^{2}V}{d\alpha^{2}}\frac{D_{1}^{2}}{c}d\alpha\ d\beta\ d\gamma$

since D₁ is independent of α. The surface-integral for the two opposite faces of the element of volume, taken with respect to the interior of that volume, will be the difference of these quantities, or

$\frac{d^{2}V}{d\alpha^{2}}\frac{D_{1}^{2}}{c}d\alpha\ d\beta\ d\gamma$

Similarly the surface-integrals for the other two pairs of forces will be

$\frac{d^{2}V}{d\beta^{2}}\frac{D_{2}^{2}}{c}d\alpha\ d\beta\ d\gamma$ and $\frac{d^{2}V}{d\gamma^{2}}\frac{D_{3}^{2}}{c}d\alpha\ d\beta\ d\gamma$

These six faces enclose an element whose volume is

$ds_{1}ds_{2}ds_{3}=\frac{D_{1}^{2}D_{2}^{2}D_{3}^{2}}{c^{3}}d\alpha\ d\beta\ d\gamma$

and if ρ is the volume-density within that element, we find by Art. 77 that the total surface-integral of the element, together with the quantity of electricity within it, multiplied by 4π is zero, or, dividing by $d\alpha\ d\beta\ d\gamma$,

$\frac{d^{2}V}{d\alpha^{2}}D_{1}^{2}+\frac{d^{2}V}{d\beta^{2}}D_{2}^{2}+\frac{d^{2}V}{d\gamma^{2}}D_{3}^{2}+4\pi\rho\frac{D_{1}^{2}D_{2}^{2}D_{3}^{2}}{c^{2}}=0,$

(11)

which is the form of Poisson’s extension of Laplace’s equation re erred to ellipsoidal coordinates.

If ρ = 0 the fourth term vanishes, and the equation is equivalent to that of Laplace.

For the general discussion of this equation the reader is referred to the work of Lamé already mentioned.

149.] To determine the quantities $\alpha,\ \beta,\ \gamma$ we may put them in the form of ordinary elliptic functions by introducing the auxiliary angles $\theta,\ \phi$ and ψ, where

$\begin{array}{lcc} \lambda_{1}=b\sin\theta, & & (12)\\ \\\lambda_{2}=\sqrt{c^{2}\sin^{2}\phi+b^{2}\cos^{2}\phi}, & & (13)\\ \\\lambda_{3}=\frac{c}{\sin\psi}. & & (14)\end{array}$

If we put b = kc, and k² + k'² = 1, we may call k and k' the two complementary moduli of the confocal system, and we find

$\alpha=\int_{0}^{\theta}\frac{d\theta}{\sqrt{1-k^{2}\sin^{2}\theta}},$

(15)

an elliptic integral of the first kind, which we may write according to the usual notation F(kθ).

In the same way we find

$\beta=\int_{0}^{\phi}\frac{d\phi}{\sqrt{1-k'^{2}\cos^{2}\phi}}=F(k')-F(k'\phi),$

(16)

where Fk' is the complete function for modulus k',

$\gamma=\int_{0}^{\psi}\frac{d\psi}{\sqrt{1-k{}^{2}\sin^{2}\psi}}=F(k\psi).$

(17)

Here α is represented as a function of the angle θ, which is a function of the parameter λ₁, β as a function of ϕ and thence of λ₂, and γ as a function of ψ and thence of λ₃.

But these angles and parameters may be considered as functions of $\alpha,\ \beta,\ \gamma$. The properties of such inverse functions, and of those connected with them, are explained in the treatise of M. Lamé on that subject.

It is easy to see that since the parameters are periodic functions of the auxiliary angles, they will be periodic functions of the quantities $\alpha,\ \beta,\ \gamma$: the periods of λ₁ and λ₃ are 4F(k) and that of λ₂ is 2F(k').

Particular Solutions.

150.] If V is a linear function of $\alpha,\ \beta$, or γ, the equation is satisfied. Hence we may deduce from the equation the distribution of electricity on any two confocal surfaces of the same family maintained at given potentials, and the potential at any point between them.

The Hyperboloids of Two Sheets.

When α is constant the corresponding surface is a hyperboloid of two sheets. Let us make the sign of a the same as that of x in the sheet under consideration. We shall thus be able to study one of these sheets at a time.

Let α₁, α₂ be the values of α corresponding to two single sheets, whether of different hyperboloids or of the same one, and let V₁, V₂ be the potentials at which they are maintained. Then, if we make

$V=\frac{\alpha_{1}V_{2}-\alpha_{2}V_{1}+\alpha\left(V_{1}-V_{2}\right)}{\alpha_{1}-\alpha_{2}},$

(18)

the conditions will be satisfied at the two surfaces and throughout the space between them. If we make V constant and equal to V₁ in the space beyond the surface a α₁ and constant and equal to V₂ in the space beyond the surface α₂, we shall have obtained the complete solution of this particular case.

The resultant force at any point of either sheet is

$R_{1}=-\frac{dV}{ds_{1}}=-\frac{dV}{d\alpha}\frac{d\alpha}{ds_{1}},$

(19)

or

$R_{1}=\frac{V_{1}-V_{2}}{\alpha_{1}-\alpha_{2}}\frac{c}{D_{2}D_{3}}$

(20)

If p₁ be the perpendicular from the centre on the tangent plane at any point, and P₁ the product of the semi-axes of the surface, then p₁D₂D₃ = P₁.

Hence we find

$R_{1}=\frac{V_{1}-V_{2}}{\alpha_{1}-\alpha_{2}}\frac{cp_{1}}{P_{1}},$

(21)

or the force at any point of the surface is proportional to the perpendicular from the centre on the tangent plane.

The surface-density σ may be found from the equation

4πσ = R1.

(22)

)

The total quantity of electricity on a segment cut off by a plane whose equation is x = a from one sheet of the hyperboloid is

$Q=\frac{c}{2}\frac{V_{1}-V_{2}}{\alpha_{1}-\alpha_{2}}\left(\frac{a}{\lambda_{1}}-1\right).$

(23)

The quantity on the whole infinite sheet is therefore infinite. The limiting forms of the surface are :—

(1) When α = F_(k) the surface is the part of the plane of xz on the positive side of the positive branch of the hyperbola whose equation is

$\frac{x^{2}}{b^{2}}-\frac{z^{2}}{c^{2}}=1.$

(24)

(2) When α = 0 the surface is the plane of yz.

(3) When α = − F_(k) the surface is the part of the plane of xz on the negative side of the negative branch of the same hyperbola.

The Hyperloloids of One Sheet.

By making β constant we obtain the equation of the hyperboloid of one sheet. The two surfaces which form the boundaries of the electric field must therefore belong to two different hyperboloids. The investigation will in other respects be the same as for the hyperboloids of two sheets, and when the difference of potentials is given the density at any point of the surface will be proportional to the perpendicular from the centre on the tangent plane, and the whole quantity on the infinite sheet will be infinite.

Limiting Forms.

(1) When β = 0 the surface is the part of the plane of xz between the two branches of the hyperbola whose equation is written above, (24).

(2) When β = F(k')) the surface is the part of the plane of xy which is on the outside of the focal ellipse whose equation is

$\frac{x^{2}}{c^{2}}+\frac{y^{2}}{c^{2}-b^{2}}=1.$

(25)

The Ellipsoids.

For any given ellipsoid γ is constant. If two ellipsoids, γ₁ and γ₂, be maintained at potentials V₁ and V₂ then, for any point γ in the space between them, we have

$V=\frac{\gamma_{1}V_{2}-\gamma_{2}V_{1}+\gamma\left(V_{1}-V_{2}\right)}{\gamma_{1}-\gamma_{2}}.$

(26)

The surface-density at any point is

$\sigma=-\frac{1}{4\pi}\frac{V_{1}-V_{2}}{\gamma_{1}-\gamma_{2}}\frac{cp_{3}}{P_{3}},$

(27)

where p₃ is the perpendicular from the centre on the tangent plane, and P₃ is the product of the semi-axes.

The whole charge of electricity on either surface is

$Q_{2}=c\frac{V_{1}-V_{2}}{\gamma_{1}-\gamma_{2}}=-Q_{1},$

(28)

a finite quantity.

When γ = F(k) the surface of the ellipsoid is at an infinite distance in all directions.

If we make V₂ = 0 and γ₂ = F(k), we find for the quantity of electricity on an ellipsoid maintained at potential V in an infinitely extended field,

$Q=c\frac{V}{F(k)-\gamma}.$

(29)

The limiting form of the ellipsoids occurs when γ = 0, in which case the surface is the part of the plane of xy within the focal ellipse, whose equation is written above, (25).

The surface-density on the elliptic plate whose equation is (25), and whose eccentricity is k, is

$\sigma=\frac{V}{2\pi\sqrt{c^{2}-b^{2}}}\frac{1}{F(k)}\frac{1}{\sqrt{1-\frac{x^{2}}{c^{2}}-\frac{y^{2}}{c^{2}-b^{2}}}},$

(30)

and its charge is

$Q=c\frac{V}{F(k)}.$

(31)

Particular Cases.

151.] If k is diminished till it becomes ultimately zero, the system of surfaces becomes transformed in the following manner :—

The real axis and one of the imaginary axes of each of the hyperboloids of two sheets are indefinitely diminished, and the surface ultimately coincides with two planes intersecting in the axis of z.

The quantity α becomes identical with θ, and the equation of the system of meridional planes to which the first system is reduced is

$\frac{x^{2}}{(\sin\alpha)^{2}}-\frac{y^{2}}{(\sin\alpha)^{2}}=0.$

(32)

The quantity β is reduced to

$\beta=\int\frac{d\phi}{\sin\phi}=\log\tan\frac{\phi}{2},$

(33)

whence we find

$\sin\phi=\frac{c}{e^{\beta}+e^{-\beta}},\ \cos\phi=\frac{e^{\beta}-e^{-\beta}}{e^{\beta}+e^{-\beta}}.$

(34)

If we call the exponential quantity $\tfrac{1}{2}\left(e^{\beta}+e^{-\beta}\right)$ the hyperbolic cosine of β, or more concisely the hypocosine of β, or cos hβ, and if we call $\tfrac{1}{2}\left(e^{\beta}-e^{-\beta}\right)$ the hyposine of β, or sin hβ, and if by the same analogy we call

	$\frac{1}{\cos h\beta}$	the hyposecant of β, or sec hβ,
	$\frac{1}{\sin h\beta}$	the hypocosecant of β, or $\operatorname{cosec}\ h\beta$,
	$\frac{\sin h\beta}{\cos h\beta}$	the hypotangent of β, or tan hβ,
and	$\frac{\cos h\beta}{\sin h\beta}$	the hypocotangent of β, or cot hβ;

then $\lambda_{2}=c\sec\ h\beta$, and the equation of the system of hyperboloids of one sheet is

$\frac{x^{2}+y^{2}}{(\sec h\beta)^{2}}-\frac{z^{2}}{(\tan h\beta)^{2}}=c^{2}.$

(35)

The quantity γ is reduced to ψ, so that $\lambda_{3}=c\ \operatorname{cosec}\gamma$, and the equation of the system of ellipsoids is

$\frac{x^{2}+y^{2}}{(\sec\gamma)^{2}}+\frac{z^{2}}{(\tan\gamma)^{2}}=c^{2}.$

(36)

Ellipsoids of this kind, which are figures of revolution about their conjugate axes, are called Planetary ellipsoids.

The quantity of electricity on a planetary ellipsoid maintained at potential V in an infinite field, is

$Q=c\frac{V}{\frac{\pi}{2}-\gamma}$

(37)

where csec γ is the equatorial radius, and ctan γ is the polar radius.

If γ = 0, the figure is a circular disk of radius c, and

$\sigma=\frac{V}{\pi^{2}\sqrt{c^{2}-r^{2}}},$

(38)

$Q=c\frac{V}{\frac{\pi}{2}}.$

(39)

152.] Second Case. Let b = c, then k = 1 and k' = 0,

$\alpha=\log\tan\frac{\pi-2\theta}{4}\ \mathrm{whence}\ \lambda_{1}=c\tan h\alpha$

(40)

and the equation of the hyperboloids of revolution of two sheets becomes

$\frac{x^{2}}{(\tan h\alpha)^{2}}-\frac{y^{2}+z^{2}}{(\sec h\alpha)^{2}}=c^{2}.$

(41)

The quantity β becomes reduced to ϕ, and each of the hyperboloids of one sheet is reduced to a pair of planes intersecting in the axis of x whose equation is

$\frac{y^{2}}{(\sin\beta)^{2}}-\frac{z^{2}}{(\sin\beta)^{2}}=0.$

(42)

This is a system of meridional planes in which β is the longitude.

The quantity γ becomes $\log\tan\tfrac{\pi-2\psi}{4}$, whence λ₃ = ccot hγ,, and the equation of the family of ellipsoids is

$\frac{x^{2}}{(\cot h\gamma)^{2}}+\frac{y^{2}+z^{2}}{(\operatorname{cosec}\ h\gamma)^{2}}=c^{2}.$

(43)

These ellipsoids, in which the transverse axis is the axis of revolution, are called Ovary ellipsoids.

The quantity of electricity on an ovary ellipsoid maintained at a potential V in an infinite field is

$Q=c\frac{V}{\gamma}.$

(44)

If the polar radius is $A=c\ \cot h\gamma$, and the equatorial radius is $B=c\ \operatorname{cosec}h\gamma$,

$\gamma=\log\frac{A+\sqrt{A^{2}-B^{2}}}{2B}$

(45)

If the equatorial radius is very small compared to the polar radius, as in a wire with rounded ends,

$\gamma=\log\frac{A}{B},\ \mathrm{and}\ Q=\frac{AV}{\log A-\log B}$

(46)

When both b and c become zero, their ratio remaining finite, the system of surfaces becomes two systems of confocal cones, and a system of spherical surfaces of which the radius is inversely proportional to γ.

If the ratio of b to c is zero or unity, the system of surfaces becomes one system of meridian planes, one system of right cones having a common axis, and a system of concentric spherical surfaces of which the radius is inversely proportional to γ. This is the ordinary system of spherical polar coordinates.

Cylindric Surfaces.

153.] When c is infinite the surfaces are cylindric, the generating lines being parallel to z. One system of cylinders is elliptic, with the equation

$\frac{x^{2}}{(\cos h\alpha)^{2}}+\frac{y^{2}}{(\sin h\alpha)^{2}}=b^{2}.$

(47)

The other is hyperbolic, with the equation

$\frac{x^{2}}{(\cos\beta)^{2}}-\frac{y^{2}}{(\sin\beta)^{2}}=b^{2}.$

(48)

This system is represented in Fig. X, at the end of this volume.

Fig. X.

Confocal Paraboloids.

154.] If in the general equations we transfer the origin of co ordinates to a point on the axis of x distant t from the centre of the system, and if we substitute for x,λ,d, and c,t + x,t + λ,t + b, and t + c respectively, and then make t increase indefinitely, we obtain, in the limit, the equation of a system of paraboloids whose foci are at the points x = b and x = c,

$4(x-\lambda)+\frac{y^{2}}{\lambda-b}+\frac{z^{2}}{\lambda-c}=0.$

If the variable parameter is λ for the first system of elliptic paraboloids, μ for the hyperbolic paraboloids, and ν for the second system of elliptic paraboloids, we have λ,b,μ,c,ν in ascending order of magnitude, and

$\left.\begin{array}{rl} x= & \lambda+\mu+\nu-c-b,\\ \\y^{2}= & 4\frac{(b-\lambda)(\mu-b)(\nu-b)}{c-b},\\ \\z^{2}= & 4\frac{(c-\lambda)(c-\mu)(\nu-c)}{c-b};\end{array}\right\}$

(50)

$\left.\begin{array}{l} \lambda=\frac{1}{2}(b+c)-\frac{1}{2}(c-v)\cos h\alpha,\\ \mu=\frac{1}{2}(b+c)-\frac{1}{2}(c-b)\cos\beta,\\ \nu=\frac{1}{2}(b+c)+\frac{1}{2}(c-b)\cos h\gamma;\end{array}\right\}$

(51)

$\left.\begin{array}{l} x=\frac{1}{2}(b+c)+\frac{1}{2}(c-b)(\cos h\gamma-\cos\beta-\cos h\alpha).\\ \\y=2(c-b)\sin h\frac{\alpha}{2}\sin\frac{\beta}{2}\cos h\frac{\gamma}{2},\\ \\z=2(c-b)\cos h\frac{\alpha}{2}\cos\frac{\beta}{2}\sin h\frac{\gamma}{2}.\end{array}\right\}$

(52

When b = c we have the case of paraboloids of revolution about the axis of x, and

$\begin{array}{l} x=a\left(e^{2\alpha}-e^{2\gamma}\right),\\ y=2ae^{\alpha+\gamma}\cos\beta,\\ z=2ae^{\alpha+\gamma}\sin\beta.\end{array}$

(53)

The surfaces for which β is constant are planes through the axis, β being the angle which such a plane makes with a fixed plane through the axis.

The surfaces for which α is constant are confocal paraboloids. When α = 0 the paraboloid is reduced to a straight line terminating at the origin.

We may also find the values of $\alpha,\ \beta,\ \gamma$ in terms of r,θ and ϕ, the spherical polar coordinates referred to the focus as origin, and the axis of the parabolas as axis of the sphere,

$\begin{array}{l} \alpha=\log\left(r^{\frac{1}{2}}\cos\frac{1}{2}\theta\right),\\ \beta=\phi,\\ \gamma=\log\left(r^{\frac{1}{2}}\sin\frac{1}{2}\theta\right).\end{array}$

(54)

We may compare the case in which the potential is equal to α, with the zonal solid harmonic r_iQ_i. Both satisfy Laplace’s equation, and are homogeneous functions of x, y, z, but in the case derived from the paraboloid there is a discontinuity at the axis, and i has a value not differing by any finite quantity from zero.

The surface-density on an electrified paraboloid in an infinite field (including the case of a straight line infinite in one direction) is inversely as the distance from the focus, or, in the case of the line, from the extremity of the line.

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