31 G THE MECHANICS OF THE EARTHS ATMOSPHERE. 



X. TIDAL EBB AND FLOW OF THE ATMOSPHERE. 



In order to facilitate the comparison of the problems treated in Sec- 

 tions viii aud ix with the computations that have been made for the 

 tidal ebb and flow, I will allow myself to add some things that do not 

 properly belong to the subject of this investigation. The following 

 formulae differ from the ordinary ones only in the notation, and in tlie 

 fact that the velocities are retained in place of the displacements.* 



In the rotating spherical shell of radius 8, and of constant temperature 

 T, the attraction of the sun produces motions for which the following 

 equations, deduced from equations (7) aud (10a), hold good: 



-± — p, =~—2nc cos go 



-4r- ^=^+2 n o cos go .... 15. 



8 sin go d A J t 



A)_(& sin co ) 3 c \ 



\ ?go ~^~ n.) 



Vindicates the potential of the sun at the point (go, A) of the rotating 

 spherical shell. When the sun stands over the equator, its distance 

 from the earth being P, its mass M, the constant of attraction ?c, we 

 have then for the potential 



k M [P 2 - 2 P Ssiu go cos (nt + A) + S 2 ]~ h 



This being developed according to the powers of p we obtain at first 



terms that have no, or at least very slight, import for the tidal ebb and 

 flow ; then come those that are to be subtracted when we consider the 

 motion of the fluid as relative only to the center of gravity of the earth. 

 That part of the potential which causes the semidiurnal tide we desig- 

 nate by Tin order to substitute it in the equations (15). 



v= 3 u M S 2 sin2 w cos (2 nf + 2 A) = if (a?) cos (2 nt + 2 A) 

 4 P 3 



Put also 



6 = E(go) cos (2 nt + 2 A) 



b= (p{co) sin (2 nt + 2 A) 



c = f (go) cos (2 nt + 2 A) 



and 



H-RT • E = G(co) 



and eliminate cp, ip from equations (15) we thus obtain 



%®. sin 2 w — t^Sin go cos go + G (4 7c sin 4 go + 2 sin 2 go— 8) 

 a go 2 a go 



= 4:lcHmn i co (16) 



* Compare, for example, the concise presentation by G. H. Darwin in the Encyclo- 

 paedia Britannica, 9th edition, article "Tides." 



