Page:Scientific Memoirs, Vol. 2 (1841).djvu/242

230 magnetic fluids in the space $$S$$, the magnetic action of which, in all other spaces $$S'$$ and $$S''$$, will be exactly similar to that of the currents.

This important proposition, which has been already mentioned (Art. 3.), rests on the following grounds: first, that these currents may be resolved into an infinite number of elementary currents (i. e. such as may be considered linear); secondly, the well-known theorem, first demonstrated, I believe, by Ampère, that in place of each linear current bounding an arbitrary surface, we may substitute a distribution of the magnetic fluids on both sides of this surface, at immeasurably small distances from it, with the same action; thirdly, the evident possibility of assigning for every re-entering line inside $$S$$, a surface bounded by it and situated wholly inside $$S$$.

If we designate by $$-v$$ the aggregate of all the quotients produced by dividing all the elements of the imaginary magnetic fluid by the distance of an indeterminate point, $$O$$ in $$S'$$ or $$S''$$; of course it is understood that the elements of the southern fluid are to be considered as negative. Then will the partial differential quotients of $$v$$, (just like those of $$V$$ in our theory) express the components of the magnetic force which the galvanic currents produce at $$O$$.

Although we must defer to another opportunity the detailed developement of the theory from which the proposition employed in the last article is taken, yet there is an important point relating to it which deserves to be noticed here. If we construct two different surfaces, $$F$$ and $$F'$$, each bounded by the same linear current $$G$$,—and (taking the simplest case for the sake of brevity) having no other point in common,—they will include a portion of space. Now, if $$O$$ be situated without this space, we obtain for that constant portion of $$v$$ which belongs to $$G$$, one and the same value, whether we assign the magnetic fluids to $$F$$ or $$F'$$; and this value is equal to the product of the intensity of the galvanic current $$G$$ (measured by an appropriate unity) multiplied by the solid angle, the summit of which is at $$O$$, and which is included by straight lines, drawn from $$O$$ to the points of $$G$$; or, which is the same thing, multiplied by that portion of the spherical surface described with radius $$1$$ round $$O$$, which is the common projection of both $$F$$ and $$F'$$.