162 Dr. E. H. Barton on the 



The wavelet is obviously a circular one of' radius A'Q = vt 

 described about a centre A', distant horizontally u±t from A. 

 From B draw BQ tangential to this arc, then BQ is the 

 refracted wave-front required. And we have by the figure 



ABA . BA' B'A-AA'4B'B 

 cosec QBA' = ^ = - 



B'A Uxt — Uot 



B'C vt 



or cosec 6 X = cosec O -. .... (2) 



This result is obtained very simply by Lord Rayleigh by 

 consideration of the velocity of the trace of the wave-front 

 on the surface of separation, i. e., referring to fig. 2, the 

 velocity of the point G along AB. This velocity is seen to be 

 susceptible of two expressions according as G is treated as a 

 point on FG or as a point on GH. Rayleigh thus obtains 

 V v 



sin do sin Oi y ' 



which is identical with (2) already found by Huyghens' 

 principle. The advantage of the longer method adopted here 

 lies in its power to treat the direction of propagation also. 

 Thus, by equation (1), or by fig. 2, in which QN is perpen- 

 dicular to AB, we have 



tan 0! = tan 1 -\ — ^sec^, .... (4) 

 which completes the solution. 



Refraction through any Number of Parallel Wind-Zones. — 

 Consider, now, any number (n + 1) of horizontal zones, in 

 each of which the wind is everywhere the same and hori- 

 zontal, but let the wind-speeds in the different zones, 

 beginning from the lowest, be u , u h u 2 , . . . U*, and let 

 the angles which the wave-fronts make with the horizontal 

 and the rays with the vertical be denoted respectively by 

 and <f> with corresponding subscripts. Then from (2) we have 



cosec #! = cosec O — (u 1 — u )/v, 



cosec 2 = cosec 1 — {^—u-^/v, 



cosec n — cosec 0»_i— (i/»— t4-i)/t>. 

 Hence, on addition, we obtain 



cosec n — cosec u — - — -. . . . (5) 



Also by (1) or (4) we have for the final direction of 

 propagation 



tan<£» = tan 0„+ — sec0„. . . . . (6) 



We thus see that the final inclination of wave-front and 

 direction of propagation are each independent of the constants 



