Page:Scientific Memoirs, Vol. 2 (1841).djvu/243

If, on the other hand, $O$ be situated inside the space included by $F$ and $F'$, it is true that the two values of the part of $v$ in question will not be the same, whether we assign the magnetic fluids to $F$ or to $F'$, because different parts of the spherical surface alluded to correspond to them,—which parts, taken together, make up the whole spherical surface. But as the galvanic current has opposite directions towards $F$ and $F'$, opposite signs must be applied in the two cases to the intensity of the current, in the multiplication into the parts of the spherical surface. The consequence is, that the algebraic difference between the values of the part of $v$ in question is equal to the product of the intensity of the current multiplied by the whole spherical surface, or by $4\pi$.

Hence it may easily be deduced, that if $O$ is situated in $S''$, the value of $v$ remains independent of the choice of the connecting surface; that if, on the other hand, $O$ is situated in $S'$, the absolute value of $v$ does indeed depend on that choice, but the differential of $v$ does not.

The highly fruitful theorem here touched upon,—according to which, in relation to the magnetic action of a linear galvanic current, the product of the intensity of that current, into the portion of spherical surface which is bounded by the line of current from $O$ outwards, has the same import in regard to attracting or repelling forces, as the parts of the mass divided by the distance from $O$,—still requires in its generality many fuller explanations, which must be reserved for a detailed treatment of this subject.

39.

The value of $v$, which in general is a function of $r$, $u$, and $\lambda$, passes on the surface of the earth into a function of $u$ and $\lambda$, and

$-{\frac {d\,v}{R\,d\,u}},\quad -{\frac {d\,v}{R\sin u\,d\,\lambda }}$

are the horizontal components of the magnetic force proceeding from the galvanic currents, directed respectively towards the north and west. It is manifest that the remarkable propositions mentioned in Art. 15. and 16. hold good likewise in this case. But as to the third component, the vertical magnetic force, the case will be somewhat different, if the agents are situated above, from what it would be supposing them to be situated in the interior. To eliminate the vertical magnetic force resulting from