396 Mr. Gr. H. Livens on the Flux of 



to be a priori impossible that one of the forms under review 

 does represent the physical facts of the case. 



There is a further difficulty of a similar kind involved in the 

 general form of theory alternative to Poynting's and arising 

 from the fact that it is usually possible to determine the 

 static potential of the field only to an additive constant. 

 This means that on such a theory the energy flux vector is 

 determined only to a constant multiple of the total current. 

 But this uncertain term in all cases merely represents a 

 flux vector possessing the usual stream property of hydro- 

 dynamics, and is not therefore relevant to the theory. 

 A similar difficulty besets all theories of this kind, including 

 even Poynting's, and it is of an entirely different nature to 

 that at present under discussion. 



Although some of these criticisms may appear in them- 

 selves to be sufficiently decisive, we shall not attempt to 

 draw any definite conclusions from them until we have 

 examined the behaviour of the different forms of theory 

 when applied to definite types of electromagnetic field. 

 We shall, however, confine our discussion to Macdonald's 

 own form of the theory and the generalized form of it 

 previously obtained, and in each problem we shall choose 

 those field potentials which have already been found most 

 convenient and appropriate. 



8. We first examine the circumstances in a simple radiation 

 field in which the propagation takes place by simple harmonic 

 plane polarized waves. If the direction of propagation is 

 along the axis of z in a rectangular coordinate system and 

 the radiation is everywhere parallel to the axis of y, we may 

 take 



H x = H, = 0, H y = Am e™t-(«+^+<\ 



where %ir\n is the period of the oscillation and m the phase 

 constant. 



Under the present conditions the vector potential is 

 sufficiently defined by the relation 



dz J 



'*-- H, 



determining its single component A^ in the form 



A-^m int - (a+ib)z+i6 m 



a -{-ib 

 In this case also it is usual to assume that the scalar 



