42 



THE MECHANICS OF THE EARTH's ATMOSPHERE. 



Here, as before, a, ft, y indicate the augles between the coordinate 

 axes and the normal to the appropriate portion of the surface of Sx. 



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

 rotation. 



We now obtain the values of w, v, n-, that satisfy the equations (I4) 

 aMd (2) bv putting 



1 



u = - — _i_ - — 





dP ?L 



?Z 





dP dM ^L 



,10 ^ dx dy \ 



and determine the quantities L, 31, N, P by the conditions that within 

 the region 81 we must have 



Hl ^'^ '_^ _ o - 

 M^ +,V "^-^-^ ~"'^' 



fM fM , cT-ill 





+ 



^y' 



+ ^ 



-2v, 



(5) 





4- 



IP 



+ 



])Z' 



= 0, 



3 



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

 Jf are the potential functions of imaginary magnetic masses distributed 



-a 



■c 



through the space S\ with the densities — ^ -—!l^--i- P is the poten- 



tial lunction 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, b, c; and by ^„ , ?;<. , Co the values of ^, ?/, Z at the point a, h, c, 

 then 



L = - 



j\I=- 



1 



- da (lb de, 



1 



Va 



"'da db dc, } (5«) 



^'=-0-^ 



— da db dc. 



where the integration is extended over the space >S'i and 



da db dc, 



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

 extended over the exterior space *S'i, that includes the region S. The 



