110 Dr. H. Bateman on an 



and the field is then completely determined when the motion 

 of the electron is known. 



4. The part of the field depending on the function 6 

 represents the radiant field added by Page to allow for the 

 effect of rotation. A moving line of electric force in this 

 field is given by the equations Q— constant, r = constant. 

 When t is constant the lines of electric force associated with 

 a given value of r are cut out on the sphere (2) by the planes 

 through the polar line of a line which meets the sphere in 

 two real points A and B. The directions SA and SB are 

 "" associated " directions of projection, the points A and B 

 being collinear with the point T which would be reached by 

 S at time t if it continued to move with the velocity which it 

 has at the instant of time r. As t and r vary the points A 

 and B trace out two associated lines of electric force. These 

 may be regarded as examples of the guiding lines of force of 

 which Sir Joseph Thomson speaks. The other lines of force 

 circle round one of these lines. 



The lines. of magnetic force being orthogonal to the lines 

 of electric force are circles through pairs of points such as 

 A and B. It should be noticed that the field vectors in the 

 radiant field may also be expressed in the form 



o(y, «) '• r d{x, t) 



■) 



t) 



where 



, P (j; -g) + Q(y-,) + R( f -g)-S(t-T) . A 



<£= lo g- - U(il ,_ f;+ v(y-,)+ w(z-r> ~ = l0 «B' 

 su=V"-/"?', sv=r'x-fv, sw=f>-vx, 



A,=VR-WQ, /t=WP-UR, k=UQ-VP, 



, AB 



The lines of magnetic force are given by = constant, 

 t = constant, and are traced out by particles which travel 



A B 



along the radii for which ir/r? -tf? an d t are constant. There 



are two such radii for each set of values of </>, i/r, and r. The 

 line joining the positions at time t of the magnetic particles 

 which travel along these radii passes through a fixed point 

 Q(*, t) given by the equations A = 0, B = 0, M = 0. The 

 polar plane of Q with respect to the sphere (2) divides the 

 positive magnetic charges on the sphere from the negative 



