Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/75

second magnetic shell whose strength is φ', the values of F, G, H will be

$F=\phi '\int {{\frac {1}{r}}{\frac {dx}{ds'}}ds'},\quad G=\phi '\int {{\frac {1}{r}}{\frac {dy}{ds'}}ds'},\quad H=\phi '\int {{\frac {1}{r}}{\frac {dz}{ds'}}ds'}.$

(14)

where the integrations are extended once round the curve s', which forms the edge of this shell.

Substituting these values in the expression for M we find

$M=-\phi \phi '\iint {{\frac {1}{r}}\left({\frac {dx}{ds}}{\frac {dx}{ds'}}+{\frac {dy}{ds}}{\frac {dy}{ds'}}+{\frac {dz}{ds}}{\frac {dz}{ds'}}\right)dsds'}$

(15)

where the integration is extended once round s and once round s'. This expression gives the potential energy due to the mutual action of the two shells, and is, as it ought to be, the same when s and s' are interchanged. This expression with its sign reversed, when the strength of each shell is unity, is called the potential of the two closed curves s and s'. It is a quantity of great importance in the theory of electric currents. If we write ε for the angle between the directions of the elements ds and ds', the potential of s and s' may be written

$\iint {{\frac {\cos \epsilon }{r}}ds\,ds'}.$

(16)

It is evidently a quantity of the dimension of a line.