98 Mr. E. A. Milne on 



above principle, taken at the moment when the wave-front 

 passes through the point. 



In all the applications we shall be concerned with cases of 

 steady motion only. Suppose then that the medium is in 

 steady motion, the components of velocity at (x, y, z) being 

 u, v, tv ; and let the velocity of sound be a, a function of 

 x,y, z. Let l 9 m, n be the direction cosines of the normal 

 to the wave-surface at the point (x, y, z). The equations of 

 a sound-ray are clearly 



dx 7 , dy dz .... 



Tt =la+u, 2 --«a + «, Jt =na + w, . (1) 



where t is the time. The ray, however, cannot be imme- 

 diately deduced by integration of these equations, since 

 /, m, n are unknown save for their initial values ; it is 

 therefore necessary to find equations for the variation of 

 7, m, n along the ray. Let (#, y, z), (a + 'Ba, y + ty, z + 8z) 

 be the points in which two neighbouring sound-rays are 

 intersected by the wave-surface at time t. Subtracting- 

 pairs of equations of the type (1) we have 



^=8(to+«)=S8.^(fc. + t.). . . (2) 



Differentiating the relation 



l8x + m8y + nSz=.Q, .... (3) 



with respect to the time, and substituting from (2) for 

 d(8x)/dt, &c, we have after reduction 



\dt ox ox ox ox/ v 



Equation (4) holds for all values of 8x : By : 8z satisfying 

 (3), hence 



1/dl oa -."du ov ow\ 



7 IT* + -s - +/ ^~ + m ^~ + n ^~~ ) 

 I \at ox ox ox ox J 



_\(dm ~da ,|Bm" "dv dw\ 



~ 771 \ d* By By By ?2 By / 



1/dn "da ou ov ow\ /er . 



which aro the equations sought. The six equations (1) and 

 (5) give the ray completely. We may express (5) more 

 simply as follows. Put 



V = a + lu + mv + nw 



