Excited Vibrations of an Atmosphere. 149 
the irrotational condition (for example, if = u = v = w ini- 
tially) they will continue to do so throughout the motion. 
Equation (6) may also be put into another form, which we 
shall find convenient. For if y is the ratio of the two specific 
heats of the gas, it can he shown that 
^ZT-yU +p l)t> ' ' - (8) 
where 'dR/'dt is the value at any point of the rate at which 
heat is being received per unit mass. 
Using (8), we have in place of (6) 
Consider next an atmosphere such as Lord Rayleigh has 
discussed in his paper already referred to. This atmosphere 
is bounded below by a plane, which we take as the plane of 
xy, while in all other directions it is unlimited. The acceler- 
ation g due to gravity acts everywhere perpendicularly to 
this plane (that is parallel to the axis of z) and is of uniform 
intensity. 
In the undisturbed state the density p is given by 
_9_ z _g_ z 
p = p e a ' 2 , while p=p e a ' 2 ; . . . (10) 
the " undisturbed " temperature T being everywhere uniform. 
Let the given thermal conditions which cause the disturbance 
be expressed by 
8T=/fc«*— "*>; (11) 
so that at each instant every plane parallel to yz is an iso- 
thermal surface ; the additional temperature-distribution $T 
being made to travel through the atmosphere in the direction 
of increasing x, like a progressive wave-train. The disturb- 
ance of pressure due to ST will evidently be of the form 
Bp = ^e^ kx - nt \ (12) 
57 being a function of z. 
From (6), remembering that 
dV , k ., c'dBp Sp 
s — = + ( h while tr ~- = -~- 
OZ c O: 0~ 
(because T and ST are independent of z), we obtain 
V%+'',.^-I^ = - R ?^. . (13) 
1 a- oz cr at" a- Qt" 
