492 WAVES OF EXPANSION. [CHAP. X 



The case that lends itself most readily to mathematical treat 

 ment is where the equilibrium-temperature is uniform*, and 

 the expansions and contractions are assumed to follow the iso 

 thermal law, so that 



c denoting the Newtonian velocity of sound. If we write 



p = p (l+s), p=p (l+s), 

 the equations (2) reduce to the forms 



du d , _ x 



dv d , _, 



*=- c V~ s) 

 * 



^ 



(4) 

 _-*.-/. 



where 



e (V/^2 /K\ 



s- M/C (&;, 



that is, 5 denotes the equilibrium- value of the condensation due 

 to the disturbing-potential H . 



The general equation of continuity, Art. 8 (4), gives, with the 

 same approximation, 



ds _ d d , . d 



We find, by elimination of it, v, w between (5) and (6), 



d 2 s 2 _ 2 . , c 2 (dp d dp d dp d\, _. ,^. 

 d& ~ p \dx dx dy dy dz dz) * 



272. If we neglect the curvature of the earth, and suppose the 

 axis of z to be drawn vertically upwards, p will be a function of z 

 only, determined by 



^? = - (1) 



On the present hypothesis of uniform temperature, we have, by 

 Boyle s Law, 



* The motion is in this case irrotational, and might have been investigated in 

 terms of the velocity-potential. 



