closed curve the extremity of the perpendicular will trace a second closed curve. Let the length of the second closed curve be σ, then the solid angle subtended by the first closed curve is
This follows from the well-known theorem that the area of a closed curve on a sphere of unit radius, together with the circumference of the polar curve, is numerically equal to the circumference of a great circle of the sphere.
This construction is sometimes convenient for calculating the solid angle subtended by a rectilinear figure. For our own purpose, which is to form clear ideas of physical phenomena, the following method is to be preferred, as it employs no constructions which do not flow from the physical data of the problem.
419.] A closed curve s is given in space, and we have to find the solid angle subtended by s at a given point P.
If we consider the solid angle as the potential of a magnetic shell of unit strength whose edge coincides with the closed curve, we must define it as the work done by a unit magnetic pole against the magnetic force while it moves from an infinite distance to the point P. Hence, if σ is the path of the pole as it approaches the point P, the potential must be the result of a line- integration along this path. It must also be the result of a line-integration along the closed curve s. The proper form of the expression for the solid angle must therefore be that of a double integration with respect to the two curves s and σ.
When P is at an infinite distance, the solid angle is evidently zero. As the point P approaches, the closed curve, as seen from the moving point, appears to open out, and the whole solid angle may be conceived to be generated by the apparent motion of the different elements of the closed curve as the moving point approaches.
As the point P moves from P to P' over the element dσ, the element QQ' of the closed curve, which we denote by ds, will change its position relatively to P, and the line on the unit sphere corresponding to QQ' will sweep over an area on the spherical surface, which we may write
| (1) |
To find Π let us suppose P fixed while the closed curve is moved parallel to itself through a distance dσ equal to PP' but in the opposite direction. The relative motion of the point P will be the same as in the real case.
