182 THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



component velocities. The solution will be quite sin]])le when Ti is 

 <leveloped into a series of spherical harmonics. 

 If we put 



and for breritv 



/3=aGR\ 



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

 stant then the solutions of the tirst two vSystems of equations are as 

 follows : 



'< t .'y .11/ t 



(4) 



In this U and F are 1 unctions of r only, and must satisfy the ditler- 

 eutial equations 



fd^F 2 dF. fj,^Q(\-ydF dEx C 



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 



H^ { ^x ,\r ^11 } 



(rv2= — — JA 



(G) 



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

 ential equations 



fdy 2dJ\^ -,dJ ^_Q-^Q(E-b) 1 



