290 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



fore the pressure po at the surface remaiuiug unchanged. If the dashes 

 relate to the second state of things, we have 



—gz —gz 



—<jz -gj^ 



p=Poe '*', p'=p^ e '*", 

 while 



rt2 o-n = a'2(3rV 



If a''i _ a^ = f5a2, we may write approximately 



p'-p _6a:'(jz zo^' 

 Po a a^ 



The alteration of pressure vanishes when z=0, and also when ^=qo. 



The maximum occurs when ^-,=1, that is, when »=— . But {p' — po) 



increases relatively to t, continually with c. 

 Again, if p denote the proportional variation of density, 



If a'-'>a-, ft is negative when 2; = 0, and becomes + cc when « = go. The 

 transition p = occurs when -^- = 1, that is, at the same place where 



p' — p reaches a maximum. 



In considering the small vibrations, the component velocities at any 

 point are denoted by w, v, ?<', the original density a becomes [ff + (T/j), 

 and the increment of pressure is dp. On neglecting the squares of 

 small quantities the equation of continuity is 



or by (3) 



dp du , di' , die d<r 

 dt dx dy dz dz 



dp du dv dw gii' t. /r;v 



di + dx+dy-^dz-^=^ ^^> 



The dynamical equations are 



ddp_ ^ du d6p_ ^dv ddp _ ^^ ^dw 

 dx 



or by (3) since 



J^-=-^W W=-^r7t' -W = -^''^'-''df 



dp = a'ffft, 



dp du „dp dv ulp dw ,n. 



""■ dx=- df' ''Ty=-dt' ''Tz=-1U • • • • W 



