326 Dr. Pocklington on the Fundamental Equations of 



VI. Experimental Fact. When the electric force is 

 independent of the time its convergence is zero. — For in all 

 that follows we only consider points in the free aether, and 

 .assume that the aether itself cannot be electrified, 



3. Let the electric force at any point be cr, where cr is a 

 function of the time t and of the vector-coordinate p of the 

 point. By II. a satisfies differential equations, which, since 

 d(dx-= — SzVj can be expressed in terms oidjdt and V« The 

 resulting" equations can by I. contain no constant vector, 

 and by III. and I. they are linear and with constant scalar 

 coefficients. If such an equation is scalar, it can only be of 

 the form 



/&• v)sv.=o, 



:and thus a must satisfy either 



S.V<r = 0or/(^, v)«r=0. 



If the latter equation is scalar, the same remark applies : and 

 hence, finally, a satisfies either S . Vo" = or a vector differ- 

 ential equation. In the latter case the equation satisfied 

 cannot reduce to a scalar equation when a is independent of 

 U Hence, by VI., cr satisfies* the equation S . \/a = 0. 



4. An equation satisfied by cr may be vector. Using the 

 formula S . V = when applied to a and its derivatives, we 

 ■get V . V — V, and by these we can reduce any function of 

 V to one containing only powers of V« To discuss the form 

 of this equation we must consider the plane wave. Let 



where j3 is a unit vector and p and q are scalars. Then 

 d/dt = ps/ — 1, V — ~<1^ — 1. Substituting in the equa- 

 tions we get Sa/3 = from that found in § 3, and from our 

 present equation we get one equation in p and q if only odd or 

 only even powers of V enter ; or, in the other case, two such 

 equations. This equation (or equations) can contain only 

 -even powers of p, as otherwise an exponential would enter 

 into the final formula, thus contradicting V. The equation, 

 which is therefore an equation in p 2 , must give only positive 

 values for p 2 , for the same reason. It can only give one 

 value, otherwise there would be more than one velocity for 



* It may be objected that in the case of the motion of a compressible 

 fluid, an equation of steady motion is S.V""=0, where <r is the vector 

 velocity of the fluid at any point, whilst this equation does not hold if 

 the motion is not steady. Here, however, the motion is not given by 

 linear equations. 



