42 



THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



Here, as before, a, /?, y indicate the angles between the coordinate 

 axes and the normal to the appropriate portion of the surface of S u 



6 indicates the angle between the normal and the resulting axis of 

 rotation. 



We now obtain the values of u, v, w, that satisfy the equations (1 4 ) 

 and (2) by putting 



u = 



dP . ^N__M -) 



dx + dy 



dy ^ dz 



dz 



dx 



(4) 



dP dM 



w —— — u- 



dz T dx dy \ 



and determine the quantities L, M, N, P by the conditions that within 

 the region Si we must have 



dx 2 + d!J 2 + dz* ~**\ 



fM tfM. d 2 M 

 dx'' + dy* dz 1 



H 



d 2 N ?N $N or 



> 



d 2 P 



+ 



$p 



+ 



dz* 

 tfP 



= 0, 



(5) 



3 



dx? t dy 2 T dz 2 



The method of integrating these last equations is well known, L, M, 

 N are the potential functions of imaginary maguetic masses distributed 



through the space Si with the densities ~- ^5 Zji • P is the poten- 



2 n' 2 7t 2 7t 7 



tial function for masses that lie outside of the region 8. If we indicate 

 by r the distance from the point x, y, z to the point whose coordinates 

 are a, 6, c; and by % a , tj a , C a the values of £, ?;, C at the point a, b, c, 

 then 



L = 



M = 



N = 



- <Za tf& dc, 



Va 



-^da db dc, J> (5a) 



C, 



aa di tfc, 



5 



where the integration is extended over the space Si and 



- da db dc, 



P = 



where Jc is an arbitrary function of a, b, c and the integration is to be 

 extended over the exterior space Si, that includes the region 8. The 



