414 MAGNETIC CONDITION OF THE GLOBE. 



We have, moreover, 



When the point P is at the surface of the Earth at P', we must 

 take r=a, and equations (4) give 



3V 



sin u = - . 

 ol 



Since, further, we have 



we get 



<)X 3 (Y sin?/) 

 1)7 ~ ~~^u 



and, therefore, 



For u = that is to say, at the north pole we have Y sin u = 0, 

 and therefore/ (/) = 0. We get then, finally, 



(6) 



We are thus led to the remarkable theorem of Gauss : 

 If for all points of the surface of the earth we know the horizontal 

 component directed towards the north, that is sufficient to give us the 

 horizontal component- turned towards the west, and therefore the total 

 horizontal component. 



