148 Dr. C. V. Burton on the Thermally 
Again, 
a/ " p a* + ^ 
of L o.'' o// o~ J 
To a first order also, when V is the gravitation potential *, 
that is (since p is not a function of t) 
From (4), (5'), and (1) we obtain 
provided that v 2 V=0. 
When there are no impressed forces capable of influencing 
the motion, the last principal term on the right hand of (6) 
disappears. When in addition there are no changes of tem- 
perature, the first term on the right hand also vanishes and 
(6) reduces to the well-known form appropriate to small 
disturbances in a gas obeying Boyle's law. 
When no forces act on the gas we have from (5), to a first 
order, 
so that, if Au, . . . are increments corresponding to a finite 
lapse of time, 
(A ! ,,A,,A,)=-J(| ( ,| / ,|)jfy^ ; . . (7) 
p being by definition independent of t. If we are dealing 
with gas which is uniform in the undisturbed state, p is every- 
where and always constant, and (7) shows that, in this case 
to our order of approximation, if w, v, w at any time satisfy 
* "Defined, for -any given point, as the work to be done against gravi- 
tative forces in bringing- unit mass from infinity up to that point. 
