218 DYNAMICAL THEOEY OF SOUND 



or c sec -Jr = const., ..................... (6) 



which is the law in question. Conversely, from (6) we can 

 derive the formula (3). When c is known as a function of y 

 the equation (6) determines the paths. 



The simplest hypothesis is that the temperature decreases 

 (or increases) upwards with a uniform gradient. This includes 

 the particular case of an atmosphere in " convective equili- 

 brium " under gravity, where the gradient is 



- 7 " 1 - (7) 



'IP 



H being the height of the homogeneous atmosphere ( 59) 

 corresponding to the temperature 6*. This is at the rate of 

 about 1C. per 100 metres. If a law of uniform decrease were 

 to hold without limitation, we should at a certain altitude 

 meet with a zero temperature (absolute). If for a moment 

 we take the origin at this level, and draw the axis of y 

 downwards, the temperature will be proportional to y t and the 



wave- velocity c to y?. Hence by (6) we have, along any ray, 



...................... (8) 



The paths are therefore cycloids, the generating circles of which 

 roll on the under side of the line y = 0. If on the other hand 

 the temperature increases upwards with a uniform gradient, the 

 paths are the cycloids whose generating circles roll on the upper 

 side of the line which corresponds to the zero of temperature. 

 In any practical case we are concerned only with the portions 

 of the curves near the vertices. The arcs may therefore be 

 taken to be circular, with a radius double the distance below 

 (or above) the level of zero temperature. In the extreme case 

 of upward diminution to which the formula (7) refers, this 

 radius will therefore be (roughly) 2 x 273 x 100 = 54600 metres, 

 for a temperature of C. 



* It was pointed out by Lord Kelvin (1862) that this is the condition into 

 which the atmosphere would be brought by the free play of convection currents 

 alone, without conduction or radiation. It is therefore one of neutral equilibrium. 

 If the temperature diminish upwards at a greater rate the equilibrium becomes 

 unstable. 



