Page:A Treatise on Electricity and Magnetism - Volume 1.djvu/224
The denominator of this fraction is the product of the squares of the semi-axes of the surface .
If we put
and if we make , then
It is easy to see that and are the semi-axes of the central section of which is conjugate to the diameter passing through the given point, and that is parallel to , and to .
If we also substitute for the three parameters their values in terms of three functions defined by the equations
148.] Now let be the potential at any point , then the resultant force in the direction of is
Since , and are at right angles to each other, the surface-integral over the element of area is
Now consider the element of volume intercepted between the surfaces , and . 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 by the surfaces and , and .