150 Dr. C. V. Burton on the Thermally 
Substituting in this from (11) and (12), and using (10), 
a solution of which is 
2 _£? 
a 2 
or 
*=,?dfo-' • • • < i0 > 
This solution corresponds to the forced vibrations, arising 
from the temperature-changes expressed by oT. The most 
general solution of (14) is found by adding to (15) the solution 
of (14) with its right-hand member replaced by zero. Such 
an addition would represent free vibrations, and it may be 
remarked in passing that the equation formed by equating to 
zero the left-hand member of (14) is equivalent to (27) of 
Lord Rayleigh's paper. It should, however, be remembered 
that the quantity there denoted by p would in our notation 
be Bp/p. 
By means of (5), bearing in mind the periodic character 
of the motion, it is easily verified that 
u= f^ 2 e i{hB -^v=0,w=0. . . (16) 
Since ro vanishes thronghont, it follows that the motion 
would not be modified (fluid friction apart) if any number of 
constraints were introduced in the form of fixed rigid hori- 
zontal planes. For these thermally excited vibrations, then, 
the solution has the same form whether Ave suppose the 
atmosphere unlimited upward, or whether we consider a thin 
lamina of air between two neighbouring horizontal planes. 
As Lord Rayleigh has done in the case of free vibrations*, 
and with much the same degree of plausibility, we may pass 
from the consideration of a thin flat lamina of air to that 
of a thin spherical sheet. In view of what has just been 
said concerning the simpler problem, it seems that the 
thermally-excited vibrations of such a sheet of air should 
not differ widely from those of an outwardly unlimited 
atmosphere. The object here is to obtain some analogy to 
the barometric variations which arise from diurnal heating 
and cooling. The problem considered is not that of the- 
earth's atmosphere, but it may be hoped that it is not too 
remote from actual conditions to afford some light thereon. 
* Loc. cit. 
