Convection Currents in the Atmosphere. 123 



All approximations hitherto made hold when the period is 

 a year or shorter. Even for a daily period we see at once 

 that the only considerable terms are given by 



P 



'~=-^-t{?+ (£+£)•}•■ ^ 



We likewise want to know the variation in the mass of 

 air per unit area of the surface. The total surface density 

 is given by 



-f 



pdz, 



«- o 

 and therefore its variable part 



a'= —ap \ Vdz 

 Jo 



= -po*?* sin-y- 4 - + - k . . . (20) 



Hence p'=ga', provided we can neglect — in comparison 



with g. For the yearly terms — is of the order of 6 x 10~ 6 , 



while for the daily terms it is about 1 cm./(l sec.) 2 . The 

 assumption made by meteorologists ih&t p' =ga' is therefore 

 justified in both cases. The result seems somewhat sur- 

 prising at first sight, since the last term neglected has 

 a factor almost equal to the radius of the earth in the 

 numerator. 



nrx 

 As the result holds for the term in V involving sin -p, we 



see that it will also hold for cos -j- , by a simple change of 



origin; further, by writing Z/(2>'+l) and l/2r respectively 

 for Z, we see that, provided r is not very great, the result 

 will hold for the terms in sin (2r + Vjttx/I and cos2r7raj/l. 

 Thus, for every term separately in the Fourier expansions of 

 the quantities concerned, p' is equal to ga' . This relation 

 therefore holds for the sum of the series. 



III. Case of Circular Symmetry. 



In this case the distribution of land and sea is supposed to 

 be symmetrical about a vertical axis, but the circumstances 

 are otherwise as in the last problem considered. Let r be 

 the distance of any point from this axis, z its height above 



