WILSON AND LEWIS. — RELATIVITY. 477 



These ait- the equations for the field of an aeeeleratetl ekK'tron 

 which were obtuiiied by Ahraluun and Sehwarzscliild.*^ It will be 

 convenient to divide the fi(>ld M into that part M' which is duv to 

 acceleration alone and that M' which is independent of acceleration. 

 The former, which is the first term in any of the above expressions 

 for M, (101)-(103), is a singular vector field, and is the only one which 

 is important at great distances from the electron, for it varies as 1/li 

 (since 1 varies with 70 whereas M'' varies as \/R-. If we divide the 

 field M' into its two parts M' = E' + H', we see here also that 



H' = Ihxe', E' = — e'xk4; (108) 



and since, in this case, l„'e' = (as may be seen by performing the 

 multiphcation) and 1. is perpendicular to e', we find that E', H' are 

 equal in magnitude. Moreover e', h' are equal in magnitude and 

 perpendicular to each other and to 1,. In other words in a radiation 

 field the electric and magnetic forces are equal in magnitude, perpen- 

 dicular to each other, and perpendicular to the "direction of propa- 

 gation." All these results are geometric consequences of the fact 

 that the 2-vector M' is singular. 



51. In four dimensional space every singular 2-vector determines 

 a singular l-^•ector, namely, a vector pointing outward along the 

 element of tangency of the 2-vector with a forward hypercone. This 

 1-vector is the complement of the 2-vector in the tangent planoid. 

 If r is the 1 -vector thus determined by the 2-vector M', then we may 

 write 



M' = uxl', 



■ where U is any unit vector in the plane of M', ])rovided the sign of U 

 be properly chosen.*^ In the case of the singular vector M' which we 

 have obtained in the previous section we may write, from (94), 



M' = -^3lx[l.(wxc)] = -|alx^*^^^-, (109) 



where a is the magnitude of l'(wxc) and therefore the last vector is 

 a unit vector. Hence we may write at once for the 1-vector deter- 

 minetl by M', 



I'= ^'^al. ■ (110) 



45 See Abraham, Theoi'ie der Elcktrizitiit, 2, p. 9.5. 



46 Owing to the nature of the geometry in a singiihir plane, tlio unit vector U 

 drawn from a given point always terminates on a definite singular line and 

 thus determines the same 2-vector uxl' for all values of U. (§31) 



