182 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



component velocities. The solution will be quite simple when T 2 is 

 developed into a series of spherical harmonics. 

 If we put 



and for brevity 



/3=aGR z , 



and indicate by Q any term of the series with its corresponding con- 

 stant then the solutions of the first two systems of equations are as 

 follows : 



x ' dy dy S \ ^ 



h\ dz dz * 



In this E and F are functions of r only, and must satisfy the differ- 

 ential equations 



f&F,2_dF\ dQ, 2 cW lQ = M(_l, a \ 

 \dr 2 r dr J dr dr t ")r 2 Jr\ r 



fd?F 2 dF\ Q iQ( dF.dE<_ ft 



\dr* + r ~drj ^^JrK^dr J 



(5) 



The constant a must be added in order to obtain the number of con- 

 stants needed in the consideration of the boundary conditions. The 

 terms depending upon the earth's rotation are 



x 2 » V tit 3y J dx s 



h* \ M- dx dV i 



H l dZ 



c 2 v 2 =^JE 

 u 



(6) 



Here also J and H are functions of r only, and must satisfy the differ- 

 ential equations 



dr^r drjjr + ^ Tr"^?^ f 



dlff, 2 dH\ „ n dff M) f ' ' ' (7) 



