100 Mr. E. A. Milne on 



which is the general condition which the function must 

 satisfy. By means of this equation it can be shown that 

 the principle assumed in § 2 is consistent with Huyghens's 

 principle, i. e. y that the positions of the wave-fronts given as 

 solutions of (1) and (5) are the envelopes of the wavelets 

 originating from various points on any one of the wave- 

 fronts, if these wavelets are also calculated by (1) and (5). 

 It must be observed, however, that the principle we have 

 assumed is really more general than Huyghens's principle 

 as usually stated, since when the medium is in non-uniform 

 motion the wavelets do not remain spheres. 



§ 4. Stratified media. The general refraction formula. 



In the applications we are concerned with in this paper 

 the medium is always taken to be a stratified one, i. e., it is 

 such that the direction of the £-axis can always be chosen so 

 that w = and u, v, a are functions of z only. In this case 

 the equations reduce to 



dx , dy dz 

 3r =to + «, ^ma+v, -=na, 



■ (12) 



ldl 1 dm 1/dn BV\ dV 



I dt ~ m dt ~ n \dt 'dz)~ n "dz' 



• (13) 



V = a + lu + mv. 





The first of equations (13) integrates immediately in the 

 form 



Z/m = constant, . . . ... . (14) 



so that the projections on Oxy of the normals to the wave- 

 front along any ray remain fixed in direction : in a 

 stratified medium the normals to the wave-fronts along a ray 

 remain parallel to a fixed plane which is perpendicular to the 

 planes of stratification ; this is also true if to is not zero 

 provided it is a function of z only. 



Choose now the axes Ox, Oy so that for a particular ray 

 the projections of the normals are parallel to Ox. Theii 



w = always, 



and so 





dz dl 1 Cfta , -."du 



whence 



dl I /"da ; d?A 



. or 



d /'a\ du 

 ~ dz\l) ~ dz ' 



)• 



