290 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



fore the pressure p at the surface remaining unchanged. If the dashes 

 relate to the second state of things, we have 



— gz ~9± 



—gz —gz 



V 



=p e a '\ p'—po <' °'\ 



while 



a 2 G a ~a n 6' Q . 



If a n _ a 2 _ ^ we may wr ite approximately 



p'—p 8 a 2 qz zJF 

 p a 1 a 2 



The alteration of pressure vanishes when s=0, and also when z=<x. 



The maximum occurs when ^==1, that is, when »=—. But (»' — » ) 



a e 



increases relatively to <r, continually with z. 

 Again, if p denote the proportional variation of density, 





If a' 2 >a 2 , p is negative when z — 0, and becomes + go when z = go. The 



transition p = 0occurs when «L = 1, that is, at the same place where 



a 2 



p' — p reaches a maximum. 



In considering the small vibrations, the component velocities at any 



point are denoted by u, v, w, the original density g becomes (ff -j- ffp), 



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



small quantities the equation of continuity is 



dp , du , dv die da 



V 



or by (3) 



ff dt +G d-*+%+ ff dz + W dz== (J 



dp du dv die 0w_ n .p.. 



di + ~dx + dlj + ~dz ~ ¥~" (5) 



The dynamical equations are 



ddp du ddp dv ddp dw 



dx = ~ ff v -d^=- ff rv -&=-'*?-*-&'> 



or by (3) since 



6p = a?ffp, 



2 dp_ du Ap dv .dp dw ,„, 



*T*—W Ty=-W dz = ~~dt ' ' ' ' () 



