414 Dr. H. Bateman on some 



permanent incidence between two lines of magnetic force in 

 which the point of intersection moves along the curve C. 



In practice it is frequently easier to ascertain whether 

 there is a relation between V, T ; V , T than whether there 

 is a relation between X, Y ; X , Y . 



Let us consider, for instance, the fields of: two moving- 

 point charges in free aether. The exact expressions for 

 X, Y ; X , Y have not been found in the general case, 

 although differential equations for finding them are known. 

 Appropriate expressions for V and T have, however, been 

 given by Mr. R. Hargreaves *, and with their aid the present 

 problem can be solved. 



Let the motion of the first point charge be specified by the 

 equations 



x = fOt)i y = v(r), z = j(t), 



then T = t, where r is given by the conditions 



[.v-Z(T)Y + [y-v(j)?+[z-ZW=<?(.t-Ty, t<*. (io) 



Let us write 

 then an appropriate value of V is f 



~~ xg+yrf + zg-cPtT'-O' ' 



where the primes denote differentiations with respect to t. 

 We shall use the suffix to distinguish the corresponding 

 quantities in the second field. We then have the three 

 equations 



-cH[t' / -Yt']-[6"-V6'] =0, 



*[fi' , -7ofo / ]+y[tf / -- ! ^o']+«[?o , '-Voro'] I nn 



-^[to"-VoTo , ]-[C-Vo^o']=0, f 



-[0-0o]=O,j 



the last one being obtained from the two equations of 

 type (10) by subtraction. 



When t, t , V, and V are given, it is generally possible to 

 solve the last three equations and (10) for a?,y, z, t. In some 

 cases, however, the equations are satisfied by an infinite number 



* Proc. Camb. Phil. Soc. 1915. 



f The charge has been assumed for simplicity to be 4?r. 



