74 PKrf, A. K H. Love. The Advancing Front of the [May 9, 



or if the damping due to radiation were very slight. The actual 

 damping of Hertz's oscillator has been investigated experimentally by 

 V. Bjerknes,* and shown to be very considerable. Accordingly, we 

 ought to take for ^ a function of the form 



(3), 



where A. is the wave-length, A a constant depending upon the amplitude 

 of the vibrations, f. a constant expressing the phase, and v a constant 

 expressing the damping. According to the experiments of Bjerknes 

 already cited, v may be taken to be about CK when the. wave-length X 

 is about 10 m. The effect of the introduction of the exponential 

 factor into the expression for ^ has been investigated in an elaborate 

 memoir by K. Pearson and A. Lee.t In that memoir it is supposed 

 that the fixed epoch from which time is measured is the instant 

 at which the vibrations begin, so that, at any instant, the field 

 expressed by (2) and (3) is confined to the region within the sphere 

 r = ct. In the expression for ^ the phase-constant e is omitted by 

 these authors. They have thus tacitly assumed that $ vanishes at the 

 front of the advancing wave. 



This front is a moving surface which is a surface of discontinuity in 

 regard to the electric and magnetic forces. Within the surface these 

 forces are expressed by the formulae already written down ; outside 

 the surface they must be expressed by some other formulse. The 

 waves, in fact, advance either through a pre-established electrostatic or 

 electromagnetic field of some kind, or possibly through a region of 

 space in which there is no electric or magnetic force. Whatever view 

 may be taken of the nature of the field outside the wave-front, definite 

 conditions must be satisfied at this surface. These conditions are 

 known, but they have not been applied to the problem in hand. It 

 seems worth while to make this application, and, in particular, to 

 ascertain the effect of these conditions in modifying the results obtained 

 by Pearson and Lee. 



Let 2 denote in general a moving surface which separates two 



Ktromagnetic fields. Then it is known that 2 moves normally to 



f with the velocity c. Let (X , Y , Z ) and (* ft, 7o ) denote the 



ctnc and magnetic forces on that side of 2 towards which 2 advances 



denote the direction cosines of the normal to 2 drawn 



this side, and let (X, Y, Z) and (a, ft y ) denote the electric 



magnetic forces on the other side. Then at any point of 2 it is 



the following six equations must be satisfied :- 



1 Ann. Phys. Chem.' (Wiedemann), vol. 44 (1891). 

 Phil. Trans./ A, vol. 1Q3, 1900. 



