652.] MAGNETIC POTENTIAL. 261
It follows from this that if fa and \jr^ are conjugate functions (Art. 183) of $ and \j/ t the curves fa may be stream-lines in the sheet for which the curves x//-/ are the corresponding- equipotential lines. One case, of course, is that in which fa = \j/ and \j/i= fa In this case the equipotential lines become current-lines, and the current-lines equipotential lines *.
If we have obtained the solution of the distribution of electric currents in a uniform sheet of any form for any particular case, we may deduce the distribution in any other case by a proper trans formation of the conjugate functions, according to the method given in Art. 190.
652.] We have next to determine the magnetic action of a current-sheet in which the current is entirely confined to the sheet, there being no electrodes to convey the current to or from the sheet.
In this case the current-function $ has a determinate value at every point, and the stream-lines are closed curves which do not intersect each other, though any one stream-line may intersect itself.
Consider the annular portion of the sheet between the stream lines $ and </>-f8(. This part of the sheet is a conducting circuit in which a current of strength 8 $ circulates in the positive direction round that part of the sheet for which $ is greater than the given value. The magnetic effect of this circuit is the same as that of a magnetic shell of strength 5 $ at any point not included in the substance of the shell. Let us suppose that the shell coincides with that part of the current-sheet for which $ has a greater value than it has at the given stream-line.
By drawing all the successive stream-lines, beginning with that for which $ has the greatest value, and ending with that for which its value is least, we shall divide the current-sheet into a series of circuits. Substituting for each circuit its corresponding mag netic shell, we find that the magnetic effect of the current-sheet at any point not included in the thickness of the sheet is the same as that of a complex magnetic shell, whose strength at any point is C-f <, where C is a constant.
If the current-sheet is bounded, then we must make C 4- <j> = at the bounding curve. If the sheet forms a closed or an infinite surface, there is nothing to determine the value of the constant C.
- See Thomson, Canib. and Dub. Math. Journ., vol.iii. p. 286.