124 Mr. H. Jeffreys on Periodic 



sea-level, and $ the azimuthal angle. Then it is obvious 

 from symmetry that if the velocity at any point be resolved 

 into three components, corresponding to r, $>, and z respec- 

 tively, the cj> component is always zero, while the other 

 two, with the pressure and the temperature, are independent 



of (f>. 



Now, as before, for a periodic variation the variable part 

 of the radiation absorbed per unit heat capacity is of the 

 form Qe~ vz , where Q is a function of r only multiplied by 

 £*¥. Most functions of r can be expressed in the form 



/too 



\ f{\) J {\r)dX 



for all values of r ; if Q*?" 1 ?* is of this type it will be suf- 

 ficient at present to treat only the case of Q = Be*^J (\r), 

 where B is a constant. Then the temperature satisfies the 

 equation 



~-*V'V = BMJ (Xr)e-'*-. . . . (1) 



A particular solution is Y — le % y t J (Xr)e- vZ f where 



b{ty-k(ifi-\*)} = B (2) 



The complementary functions must have no singularity 

 when r = 0, and vanish when z is infinite. Further, when 

 z is zero, Y must reduce to the temperature at the surface of 

 the earth. Let this be (6 + c)^J (Xr). Then the complete 

 solution is 



Y={l)e-" z + ce- m2 )J Q (\r)e*Y t , .... (3) 



where m is the root with a positive real part of m 2 = X 2 + i<yjk. 

 By making b and c functions of X and integrating with 

 regard to X from to go , it will be possible to obtain the 

 most general solution. 



The equations of motion are as before 



— = -^- 4 k\/ 2 U + o k ^~ > 



^t p d# 3 oa 



^v 1 d// , 7V7 9 , 1 7 3^ 



— — — ^-fWi' + n/^, 



d* p dy 3 dy 



"dio Tr 1 'dp' 7 „ 1 7 ^ 



ot u po oz o dz 



W 



