104 Mr. E. A. Milne on 



Using (19) we find 



. . . (23) 







and hence 



sin 8E _ 1 f z cos 6 cos 6 cos j (0 - <9 ) W - W 

 sinE ~ZJ " sin sin f (0 + o ) «o 



ipW(,)sin^o 

 + ZJ «(*) sin 



ir z cos6> cosl(6>-6 > ) « -a 

 + Z\ sm <9 sin i(6> + (9 ) a ' ^ J 



This is the exact formula for the correction in elevation, 

 evaluated in terms of quadratures. As before, we may 

 approximate to the first order, when we obtain 



8E= W^LP^ + W s . n e a^ cotf)o _ (M) . 

 a sm u a Q a 



To make use of (21) and (25) the observer must be 

 furnished with the values of the wind and temperature at 

 all heights up to the height of the source ; from these he can 

 calculate a, and the mean wind and its components along and at 

 right angles to the observed direction. The precise evalua- 

 tion of W, &c, requires a knowledge of the height of the 

 source, which is usually not known ; but an approximate 

 estimate of this will usually be sufficient. Tests of (21) and 

 (25) were frequently made during the war in connexion 

 with the location of aircraft by sound, and the corrections 

 were usually found to be adequately given by the formula?, 

 at least when the distributions of wind and temperature were 

 sufficiently known. Formula (25) is liable to be seriously in 

 error when 6 is small ; in this case the correction SE may 

 be comparable in magnitude with 6 and E themselves. 

 But when is small no approximate formula can give &E 

 accurately (unless the precise form of the wind is given), 

 since the sound-wave, travelling almost horizontally, may 

 be on the verge of being totally reflected, and the path of 

 the sound-ray is sensitive to small changes in the wind 

 or temperature distribution. More accurate forms of (25) 

 have been investigated, suitable under certain circumstances 



* See footnote on page 103. 



