TERRESTRIAL MAGNETISM. 231 



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 ques- 

 tion 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 cur- 

 rent 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 sur- 

 face, or by 4 tt. 



Hence it may easily be deduced, that if O is situated in S", 

 the value of v remains independent of the choice of the con- 

 necting 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 at- 

 tracting or repeUing 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 A,, 

 passes on the surface of the earth into a function of u and \, and 

 dv dv 



Rdu' R sin ud\ 

 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 in- 

 terior. To eliminate the vertical magnetic force resulting from 



