KECENT PROGRESS IN" THEORETICAL PHYSICS. 217 



aud if the functions L, i¥, N. and P, be taken so as to satisfy also 

 within the larger space S', the conditions 



(5) 



d^M , d^M , d^M „ „ 

 dx" du-^ dz" 



d^N d"N d^N „ 3 



c?a;^ dy'^ dz^ 



c?a;2 <Zj/2 dz" 



The analogy of these equations (5) to Poisson's equation (C) is at 

 once apparent if 



~ 2ti^ ~ 2ti' ~ 211' 



be each taken equal to 9. L, M, N, are quantities which satisfy the 

 same equations as the vector poientials of electric currents. They stand 

 in the same relation, in vortex fluid motion, to the angular velocities 

 of the core of the vortex filament, as do in electricity the vector- 

 potentials of electric currents, to what might guardedly be called 

 the mass of the currents v/hich give rise to these potentials; thus 

 again showing the help we may derive in our notions of electrical 

 strength, mass, density, or whatever we choose to call it, by com- 

 paring the "current-penetrated space," to the core of a vortex fila- 

 ment. It moreover prominently calls our attention to what may 

 be going on in the space outside the wire, as well as in the sub- 

 stance of the wire itself. Indeed, if r be the distance of a point 

 a, h, c, from a point x, y, 0, on the axis of a vortex-filament; and if 

 icl, M'a, tvl, be the values which iv^, tv"^, iv^^ have at this point, a, h, c, 

 then we will have 



^ = ~ Y-n''- fff^dadhdc 



M = L* rrr^kdadbdc (^) 



2 n JJJ r 



^ =--^V/f—dadbdc 



* V represents the ratio 3.1416. 



