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 



P = *P (3), 



c denoting the Newtonian velocity of sound. If we write 



p = p Q (l+s), p=p (I+8), 

 the equations (2) reduce to the forms 



du _ d , _, 

 dt dx^ 



dv _ 2 d f _. 

 777 ~~ ~~ c xTT, v s ~~ s /> 



(4), 



dw d , _, 



~ji ~ c ^~\ s ~ s ) 

 dt dz^ 



where 



-- fl'/c 2 ........................... (5), 



that is, s denotes the ' equilibrium- value ' of the condensation due 

 to the disturbing-potential ft'. 



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

 same approximation, 



ds d & / \ d , 



*ar ~as ^ ~ T M ~Tz (f) w} ......... (6) - 



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



^_ C 2 V2 ^- a+ffeA + 3&A + 3&.*V,_5\ a\ 



dt*~ + p Q (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 



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

 Boyle's Law, 



(2), 



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

 terms of the velocity-potential. 



