106 THE MECHANICS OF THE EARTH's ATMOSPHERE. 



The values corresponding to these are 



sin 7;i=0 

 cos f>=cos e 

 In consequence of this tlie equation above given becomes 



i-. f = — n.r-\-\off ( f^ ) +h—2 \ | e-°\ cos^ (eg) 

 ^ " / / 1 U. cos {ahi)j 



(.r = -x) 



The first term of the right-hand member is infinite, bnt all the others 

 finite when /( is a positive quantity. Therefore at great depths the 

 value of if-2 reduces to 



!/•■>= —nh-yX 



that is to say that even there also the motion is a rectilinear flow with 

 the velocity —nhz. 



The second boundary condition which has respect to the equality of 

 pressure on both sides of the boundary surface can, however, by reason 

 of the assumptions already made, be satisfied only approximately for 

 waves of small altitude. The convergence of the series under consid- 

 eration in this case depends upon the factor e-"\ When the quantity 

 h is positive and not too small the series converges relatively rapidly 

 and we obtain for this case suflicient approximation to the true value, 

 in that in the value of the pressure as deduced from equation (3) we 

 equate to zero the terms multiplied by the first to the third j^ower of 



e-", or of . , ■ • The terms that do not contain these factors serve 



only to determine the value of the constant of integration which forms 

 the left-hand side of the equation. These terms just mentioned are 

 linear functions of cos f>, cos 2fi, cos 3H, and by equating to zero the co- 

 efficients of these three quantities we satisfy equation (3) to terms that 



contain the fourth or higher power of ?-.• But this assumption cor- 



° ^ cos hi ^ 



responds only to a single possible form of wave, not to the most general 



form. It has been chosen as an exarajile on account of the simplicity 



of computation. The three equations that we obtain in this manner 



are those given below. For brevity we have put 





s-z hi^ . 7t 

 gTX{S2 — Si) 



cos nt 



z = Z cos £ 



