70 Prof. L. Vegard : Results of Northlight Investigations 



Applied to the absorption of the atmosphere p , etc., are 

 functions of the space coordinates, in fact functions of the 

 distance R from the point to the centre of the earth . 



If we suppose the temperature on the first 10 km. from 

 the ground to be on an average —23° C, and above 10 km- 

 — 55° C, we have 



Pi =p oi ^0-694 . l(r 2 M t - e -5415 . ICT 8 M Z (R-E ) ? 



where M t - is the molecular weight of the substance i, and R a 

 the radius of the earth, or the densities are expressed by 

 functions of the form 



Pi=g%e 



The differential equation of motion of the raj, when written 

 in vector form, will be : 



dt m ^ m l J ! 



K *' 



-=rd)(^)X%^" 6iR j 

 m rv J 



where v is a unit vector in the direction of the velocity. 



A general and exact solution of these equations is a very 

 difficult problem, and has not very great value for out- 

 present purpose ; it will be sufficient to get approximate 

 solutions for special cases. 



Now the resistance force only exisis quite close to the 

 surface of the earth, and in these regions the magnetic field 

 is nearly uniform, or rather the lines of force are slightly 

 converging towards the earth. Under these conditions we 

 are able to prove certain theorems which enable us to get an 

 idea of the orbits. 



We supposed the magnetic field to be produced by a 

 single magnetic pole of mass M. Let us first consider 

 the case treated by Poincare, that there is no absorbing 

 medium. Then Poincare showed that an electric ray will 

 move in an orbit, that is a geodetic line on a cone of 

 revolution with its top at the magnetic pole. Let the 

 angle between the cone axis and one of its generatrices 

 be ©. If we let the cone roll on a plane, the line of 

 contact will for each revolution move through an angle 



cp = 2 it sin co. 



A geodetic line on the surface of the cone will be a 

 straight line on the plane, and the actual orbit in space 



