230 C. F. GAUSS ox THE GENERAL THEORY OF 



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 men- 

 tioned (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 beUeve, by Am- 

 pere, 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 Une inside S, a surface bounded 

 by it and situated wholly inside S. 



If we designate by —v the aggregate of all the quotients pro- 

 duced 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 Avill the partial differen- 

 tial quotients of v, (just hke those of Fin our theory) express 

 the components of the magnetic force which the galvanic cur- 

 rents produce at O. 



38. 



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 re- 

 lating 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 ForF'; 

 and this value is equal to the product of the intensity of the 

 galvanic current G (measured by an appropriate unity) multi- 

 plied by the soUd 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, multipUed by that portion of the 

 spherical surfai;e described with radius 1 round O, which is the 

 common projection of both F and F'. 



