120 Mr. H. Jeffreys on Periodic 



By hypothesis all the terms in the second equation are 

 zero. Put 



o/ot for D/Bt, 



and let p=p +p\ where p is the pressure in the undis- 

 turbed state. p is a function of z only. 

 Then to the first order, 



-^= L K (6) 



1 "dp _ 1 + aV "bpo 1 "dp' ... 



p 02 " /0 B^ /o oz ' 



But the hydrostatic condition is 



^ = - P oZ=-m (8) 



Hence 



- ^r = —g—gotV + - ^~ (9) 



Further, X = and Z——g. 



The equations of motion then reduce to 



g = _L!^ +iV «u + |Ag, . . (10) 

 o£ po dx 3 3#' y 



"ow T7 1 dp' , JX - 72 1 7 oS ,,,. 



Tt= 9 " y -po"dI +Ww+ 3 k *i' • < U > 



p + ^ = g =a |Z (32) 



0<^ 02 o* 



Differentiating the first with regard to x, and the second 

 with regard to z, and adding, we find 



! 4 7 / „ 7T 2 \ , « "1 7 . 7TX . 



= « —gv+ 0**71^— 72 1 + 7 >- 6«~ w sm -y- e*v* 



f 1 TTX 



— -j #m + - 7 2 > c<?~ mz sin -p 1 ?*. 



