and Nature of the Cosmic Electric Rays. 71 



would be found by winding up the plane into the form 

 of the original cone. 



We consider a ray moving very far from the pole in 

 a certain direction, and we imagine a line (CM) drawn 

 through the pole which is parallel to the direction of motion 

 of the ray. Now Poincare has shown that (CM) is on the 

 surface of the cone on which the ray is moving. If we 

 imagine the surface of the cone developed on a plane, 

 CM will be a line in this plane and the orbit another 

 line AB parallel to it (PL II. fig. 6, 1.). The original distance 

 d between these two lines at the same time represents the 

 smallest distance between the pole and the orbit, or the 

 distance from the pole to the point where the orbit turns 

 back again into space. As we see, the orbit, when it is 

 developed in the plane, is independent of the properties 

 of the electric ray, but still the actual orbit in space w ill 

 be different for rays of different properties because the 

 angle </> will vary with the ray properties. 



From the results of Poincare we find 



. a mv d 



where m, e, and v are mass, charge, and velocity of the ray 

 respectively; or with a given carrier of the ray, a given 

 distance d and a magnetic mass M, the angle (j> is pro- 

 portional to the velocity. Now the number of times the 

 orbit turns round the cone on its way from infinity and 

 back to infinity (as we easily see from fig. 6) is equal 



to - = it — -., or — other conditions being the same — the 

 <p 2 mvd ft 



number of turns is inversely proportional to the velocity. 



Fig. 6 I. and n. illustrate the orbits for two different 



values of the velocity, or two different values of the 



quantity — . It appears from the figure, that for a given 



distance d, the angle between the orbit and the magnetic 

 line of force at a given distance r from the pole will always 

 be the same whatever be the properties of the electric ray, 

 this angle ty being given by the expression 



• i d 



T r 



Consequently, if at a given distance from the pole a 

 number of electric rays of different properties start at the 

 same angle i/r, all these rays will reach the same minimum 



