Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/71

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420.]
39
GENERATION OF A SOLID ANGLE.

During this motion the element QQ' will generate an area in the form of a parallelogram whose sides are parallel and equal to QQ' and PP'. If we construct a pyramid on this parallelogram as base with its vertex at P, the solid angle of this pyramid will be the increment dω which we are in search of.

To determine the value of this solid angle, let θ and θ' be the angles which ds and dσ make with PQ respect ively, and let φ be the angle between the planes of these two angles, then the area of the projection of the parallelogram ds . dσ on a plane perpendicular to PQ or r will be



ds d\sigma \sin \theta \sin \theta' \sin\phi,\,


and since this is equal to r2dω, we find



d\omega = \Pi ds d\sigma = \frac{1}{r^2} \sin\theta \sin\theta' ds d\phi.
(2)
Hence 
\Pi = \frac{1}{r^2} \sin\theta \sin\theta' \sin\phi.
(3)


420.] We may express the angles θ, θ', and φ in terms of r, and its differential coefficients with respect to s and σ, for



\cos\theta = \frac{dr}{ds}, \quad
\cos\theta = \frac{dr}{ds\sigma}, \quad \text{and }
\sin\theta \sin\theta'\cos\phi = r \frac{d^2r}{dsd\sigma}.
(4)


We thus find the following value for Π2,



\Pi^2 = \frac{1}{r^4}
\left[1 - \left(\frac{dr}{ds}\right)^2 \right]
\left[1 - \left(\frac{dr}{d\sigma}\right)^2 \right] -
\frac{1}{r^2} \left(\frac{d^2}{dsd\sigma}\right)^2.
(5)


A third expression for Π in terms of rectangular coordinates may be deduced from the consideration that the volume of the pyramid whose solid angle is dω and whose axis is r is



\frac{1}{3} r^3 domega = \frac{1}{3} r^3 \Pi ds d\sigma.


But the volume of this pyramid may also be expressed in terms of the projections of r, ds, and dσ on the axis of x, y and z, as a determinant formed by these nine projections, of which we must take the third part. We thus find as the value of Π,



\Pi = \frac{1}{r^3}
\begin{vmatrix}
\xi - x, & \eta - y, & \zeta - z, \\
\frac{d\xi}{d\sigma}, & \frac{d\eta}{d\sigma}, & \frac{d\zeta}{d\sigma}, \\
\frac{dx}{ds}, & \frac{dy}{ds}, & \frac{dz}{ds}.
\end{vmatrix}
(6)