H. A. Bumstead — Lorentz-Fitz Gerald Hypothesis. 503 



These are also the values of the electric and magnetic forces 

 produced at points outside, by Lorentz's electron, or by any 

 charged system in which, when at rest, the charge is dis- 

 tributed with spherical symmetry and which, when in motion, 

 suffers the Lorentz-FitzGerald contraction. E is the force 

 exerted by the moving charge e, upon a unit charge which is 

 at rest at the point P. If the unit charge at P is in motion 

 with the velocity u, then the force exerted upon it, which we 

 may call (E), is 



(E)=E+uxH (3) 



where uxH represents the vector product. Thus the force 

 on a charge at rest at the point, P, is in the direction of r, but 

 this is not true in general if it is in motion. 



Let us consider first the special case when the two charges 

 have the same velocity, u=v. Let the two components of E 

 parallel and perpendicular to v be E, and E 2 , respectively. 

 The force vxH will be parallel to E 2 and in the opposite 

 direction and its magnitude will be vU.. So that the corre- 

 sponding components of (E) are 



(E), = E cos (4) 



and 

 or since 



(E) 2 = E sin 6-vR 



H = ^ 2 E sin 0, 



(E) 2 = E sin 0(1 -£ 2 ) (5) 



These are the components of the actual force on the moving 

 charge at P ; if it is of opposite sign to the charge e, the force 

 will have the direction given in tig. 2. 

 When 6 = 0, (E) 2 = and 



(E) 1 = E = V 2 6 (1 ~ A') 



which is (1 — /3 2 ) times the value of the electrostatic force when 

 the charges are at rest : this corresponds to the gravitational 

 case of p. 501 when the force was in the direction of motion. 



When ^— , (E), = 0, and 



(E) 2 =E(1-/? 2 ) =V 2 ^Vl-/3 2 



which also agrees with the corresponding case for gravitation. 

 If we apply this electromagnetic law of force to gravitation 

 we are at first sight confronted with the difficulty that the 

 magnitude of the force varies not only with the distance but 



