Vibrations of an Atmosphere. 177 



energy of the vibration is the same in every wave-length, 

 not diminishing with elevation. The viscosity of the rarefied 

 air in the upper regions would suffice to put a stop to such 

 a motion, which cannot therefore be taken to represent any- 

 thing that could actually happen. 



When 2na<g, the values of m from (13) are real, and are 

 both positive. We will suppose that ?n± is greater than 

 w 2 . If to vanish with z, we have from (14) as the expression 

 of the stationary vibration 



w=cosnt(e m ^—e m -^), _ (18) 



which shows that to is of one sign throughout. Again by (8) 



— — — > (19) 



Hence dp/dz, proportional to w, is of one sign throughout ; 

 p itself is negative for small values of z, and positive for large 

 values, vanishing once when 



e ( m] -m. 2 )z — mi f m ^ (20) 



When n is small, we have approximately 



a Tv n 



m 1 = , — 2 , m 2 =-, . . . . (21) 



a 2 g g 



so that p vanishes when 



p gzja 



. v 2 



(22) 



or by (4) when 



<r/o- = n 2 a*/g 2 (23) 



Below the point determined by (23) the variation of density 

 is of one sign and above it of the contrary sign. The inte- 

 grated variation of density, represented by j ap dz, vanishes, 

 as of course it should do. 



It may be of interest to give a numerical example of (23). 

 Let us suppose that the period is one hour, so that in C.Gr.S. 

 measure w = 2tt/3600. We take a = 33 x 10 4 , #=981. Then 



showing that even for this moderate period the change of 

 sign does not occur until a high degree of rarefaction is 

 reached. 



