412 Dr. H. Bateman on some 



experience. There are, indeed, some points in favour of 

 the hypothesis in addition to its simplicity. Let us see, for 

 instance, if the above law gives the correct motion when a 

 charged particle moves in a steady uniform magnetic field 

 in which H is parallel to the axis of z. In this case we may 

 write 



A x = — ^Hz/ + ^mcTJx, A y = iHct' + ^mcTJy, 



A z — i??26'Uz, <!> -\mc 2 , 



where U is a constant vector. When these forms for A and <X> 

 are adopted, the equations of motion take the correct form : 



dx TT 1 TT d 2 x 1 TT du 



» 3r =»H, = mU I --Hj, 1 or m _ = _-H 3F , 



dy TT 1 TT d 2 y 1 n dx 



dz dH 



m - Ti = mu z = mV z . m-ri; = (). 



at dt z 



The quantity m appearing in these equations represents 

 the mass divided by the electric charge with its sign 

 changed ; we assume here that it is a constant. 



It is clear from the above example that there is some 

 ambiguity in the choice of A and <I> when the field is 

 given, for it is easy to write down other potentials such as 

 A x =— H#, A y = 0, As = 0, <3> = 0, which will specify the 

 same electromagnetic field : consequently the hypothesis is 

 in some respects incomplete* 



§ 6. Incident Fields. — Two special electromagnetic fields 

 are said to be incident when there is permanent incidence 

 between lines of electric force of the two fields. This 

 means that if a line of force of one field intersects a 

 particular line of force of the other field at a given instant, 

 it will continue to intersect it. The condition that this may 

 be true has already been found "*. If the lines of force of 

 the two fields are given by X = const., Y = const., and 

 X = const., Y = const, respectively, the condition is that 

 the Jacobian should vanish, i. e. that 



d(X, Y; Xn, Y ) _ 

 dO, y, z, t) 



This implies that Y = const, is a consequence of X = const., 

 Y = const., X = const. It should be noticed that when this 



* l Messenger of Mathematics,' 1916. 



