Vibrations of Light with Electrical Currents. 297 



development of free electricity is possible. This result is differ- 

 ent from that which Kirchhoff has deduced from the original 

 equations (1) — that is, that in the interior of a conductor there 

 is in general free electricity ; but from the whole of the present 

 investigation it will be clear that at all events this conclusion 

 cannot be drawn with any degree of certainty. 



Thus, after it has been proved that from equations (A), which 

 embrace the laws of electrical currents that are in accordance 

 with experiment, the differential equations (B) can be deduced, 

 which show that electrical currents behave in every respect like 

 the vibrations of light, the question arises whether, on the other 

 hand, the laws of electrical currents can be deduced from the 

 known laws of light. I shall now show that this is in fact pos- 

 sible, in such a manner that equations (A) can be again deduced 

 from equations (B), provided the conditions be introduced into 

 the latter which must be fulfilled at the limit of the body, and 

 which we must know in order to deduce from the differential 

 equations such others, which in a certain sense are their integrals. 

 At the same time it will be seen that these limiting conditions 

 are just the same as those I have already found (Fogg. Ann. 

 vol. cxviii. p. 126 ; Phil. Mag. S. 4. vol. xxvi. p. 93) for the com- 

 ponents of light ; so that for this calculation we need make no 

 other assumptions than just those which the theory of light gives. 



For an element of the surface which is at right angles to the 

 axis x, I have found that the magnitudes 



du d dw du 



3 ' dy dx dx dz 



on both sides of the element are equal; from this the limiting 

 conditions for all other elements of the surface will be found, 

 because the direction of the coordinate axis is arbitrarily chosen. 

 These conditions are deduced from the differential equations of the 

 components of light, which was possible in this case, because they 

 held generally for all heterogeneous media, and they remained 

 the same even after the members from the equations (B) con- 

 taining the factor k had been added to the equations, a circum- 

 stance which is now seen to be necessary. 



For a body whose conductivity is constant, which is surrounded 

 by absolute non-conductors (no matter whether they really exist 

 or not), the above-named magnitudes become zero on the surface 

 of the body, since any electrical current is impossible in the entire 

 insulating surface which surrounds the body. 



We introduce now into equations (B), instead of u, v } w, x, y, z> 



r 

 the notation u ! , v f , «/, or 1 , ij, x l -, and first suppose t substi- 

 tuted for /, where r denotes the distance of a fixed point x y z 

 Phil Mag. S. 4. Vol. 34. No. 230. Oct. 1867. X 



