Classification of Electromagnetic, Fields. 137 



an electromagnetic field with this simple type of singularity 

 is obtained by putting 



where r, M are defined by the equations 



[«-f(T)]»+[y-,(T)]»+[*-KT)]»=l«-T)», t»T, (6) 



f(7) + f'(T) + nr)<l, . . . (7) 



M=r(T)[*-fl+v(r)[y-v]+r(T)[*-a-(*-T)- (») 



If all these conditions are satisfied, there is only one 

 value of t for each point (#, y, ~, £), and M only vanishes 

 when all the terms in (6) are zero, i. e. when #_, ?/, ~, t 

 coincides with the moving-point (£, rj, f) *. This electro- 

 magnetic field, which was discovered by Lienard, is rightly 

 regarded as being of fundamental importance in the electron 

 theory of matter, and electromagnetic fields which can be 

 obtained by superposing a number of fields of this simple 

 type are studied almost exclusively. 



For the sake of thoroughness, however, it is desirable that 

 all types of electromagnetic fields should be studied, the aim 

 being, if possible, to discover a number of fundamental types 

 from which all real electromagnetic fields can be derived by 

 superposition f. 



§ 2. Solutions analogous to Lienard's may be obtained by 

 discarding the inequality (7), or by considering complex 

 functions £, 77, f and a complex variable t, or, finally, by 

 putting 



A,=2 ± Cp, A,=S±«£>, A,=S ± i<M *=0, ( 9) 

 where the summation extends over values of u for which J 



[>-*('')] ! +[>-'K<0] ! +[-— SOOJ^* 2 , • ( 10 ) 



and M=f(u)[tf-?] + V(«)[y-'»] + f(w)|>-£]- (11) 



In the first case the singularity (f, 77, f, t) travels with a 



* For further details, see Schott's ' Electromagnetic Radiation.' 

 t Differentiations with regard to the variables .r, y, z, t are supposed 

 to he included, as well as integrations with regard to variable para- 

 meters. 



\ Any finite number of roots may be chosen and the signs in (5>) 

 distributed arbitrarily and Maxwell's equations will be satisfied. In 

 making a choice of the roots and signs we must endeavour to make the 

 components of E and H single-valued functions of .r, //, s, t. 



