416 Dr. H. Bateman on some 



a direction perpendicular to itself. In particular, the con- 

 ditions are satisfied if the two point charges move in one plane. 



If only two relations are required for the vanishing of the 

 determinant, it means that each line of magnetic force of 

 one field generally has permanent intersections with a finite 

 number of lines of magnetic force of the second field. The 

 intersections lie on a moving surface, and lines of electric 

 force of the two fields have permanent intersections on this 

 moving surface *. When the two point charges move in 

 one plane, it is evident that the plane must be the surface to 

 which we have just referred ; and if we generalize this case 

 by means of the transformations occurring in the theory of 

 relativity, it appears that the moving surface is represented 

 in S4 by the hypersphere or hyperplane on which the repre- 

 sentative curves of the two paths lie. 



One relation is sufficient for the vanishing of all the 

 determinants of the array when four equations of type 



Z£ + mr) + nf + pr + q6 + r = 0, 



hi + m iV + n £ + P\ T + QiO +^i=0, 



Zfo + m Vo + n Zo + P^o + q0 -f r = 0, 



hio + m i*7o + wi?o + Pi T o + qiT + r t = 0. 



In this case the paths of the two point charges are repre- 

 sented in S4 by two curves which lie on the same sphere or 

 plane. In the latter case the two charges lie at any instant 

 on a straight line which moves in a plane with uniform 

 velocity, retaining the same direction all the time. In 

 particular the conditions are satisfied if the two charges 

 move along the same straight line. 



When only one condition is required for the vanishing of 

 all the determinants, each line of force of one field is 

 permanently incident with co 1 lines of force of the other 

 field. There is thus complete incidence of the two fields 

 and no cutting of lines of force. 



It should be noticed that when two incident special fields 

 are superposed the total field is also special f, for we have 



(E + E\ H+H')= (EH) + (EH / ) + (E'H) + (E'H') =0. 



§ 7. A system of mutually incident special fields. — The 

 last theorem may be generalized as follows : — If a system 



* It may happen that there is a set of oo 1 lines of force such that each 

 line of force meets G ° 1 lines of force of the second field. 



■j- It is often easy to find the lines of force in the total field when the 

 lines of force of the constituents are known. The example considered in 

 § 8 will illustrate this. Two point charges moving along the same straight 

 line provide another good example. 



