Sound Waves in the Atmosphere. 103 



Equations (16), (18), and (19) show that the projection of 

 the ray on z Ox is independent of the " cross wind " W, and 

 hence SE is independent of W: thus, in calculating SE, 

 T : can be taken to be the source. 



From (17) and (18) we have 



dy W'(s) 



j~ r^r cosec 0. 



dz a{z) ' 



whence Y = — I , V cosec dz, 



Jo << 



z) 

 and therefore, E denoting the angle TjO^', 



tan $4> = Y/(Z cot E) 



1 C z W r (z) 

 = -tanE 77 l ~^/-cosec0dz. . (20) 



Given W (z) and a(z), 6 is known from (19), and the 

 integral could be evaluated by quadrature. But we may 

 approximate as follows : the wind being always small com- 

 pared with the velocity of sound, we can consider 6 to be 

 constant in (20) with a resulting second order error ; to the 

 order required here, # = # =E, and so 



£</>=- ^-sec0 o , • • , . . (21)* 



a 



approximately, where bars denote mean values with respect 

 to height. This evaluates the azimuth correction. 

 Similarly, from (16) and (18), 



dx f/) W(s) , 



= cot a ^- cosec o 



dy a{z) 



X 1 C z / WW \ 



giving cot E= 7^ = 77! (cot# -^ cosec 6)dz. 



L LJo \ a(z) I 



Subtracting cot O from each side and simplifying, we have 



sinSE_ip/ sin(fl-fl ) W(*) sin O \ , . 



sinE ~ZJ \ sin " 1 " a(,c) sin / ' ' ^"" 



* These approximate formulae were first deduced by Hill, using 1 

 slightly different analysis. The equations (20) and (24) in the text, 

 via which the formulae are here deduced, are convenient when it is 

 necessary to examine the errors of (21) and (25). 



