WILSON AND LEWIS. — RELATIVITY. 489 



then M'M = /r — ^ = 2/., where L is known as the Laf,'ranf;;ian fnnc- 

 tion, and M*M* = 2e'h. It is not snrprisinjjc that the Laf,M'anf,nan 

 funetion shonhl prove to he one of tlie fnndaniental in\ ariaiits, hnt 

 it is strange that the other invariant should be a quantity whieh has 

 not been regarded as of fundamental importance in electromagnetic 

 theory. 



Since we have obtained our 2-vector from the equation 



M = <y>im, 



we may readily evaluate <(]>xM and <C> • M. By (51) as a mathematical 

 identity we have 



OxM = Ox<>xm = 0. (141) 



By (55) 



O-M = O-(Oxm) = 0«>-ni) - «>-0)ni; 



and since we have seen that in general <[>'in = 0, and substituting 

 for <Q>"(yTn. or <C>-in from the preceding section,^ ^ we find 



O-Mi^ 4 7rq. (142) 



By (52) as a mathematical identity, 



<0>-(O-M) = 0. (143) 



By the expansion of these equations we obtain directly the familiar 

 equations of the electromagnetic field and the continuity equation 



as space. Following the method of § 38 we maj^ write M as the sum of its 

 two completel}' perpendicular parts in the form 



_ J (V(M »Mr + (M.M*.)-^ + M-M ) M + (M-M*) M* 

 ~ ' V(M.Mr- + (M-M*)--^ 



J (V(M.Mr^ +" ( M.M *)^ -M-mm - (m.m*)^m* 



Now the lines in which these two completely perpendicular planes cut the space 

 ki23 ni;iy ho found by multiplying the planes by k, by itiner multiplication. 

 As k4«M = e and k4«M* = — h, we have for the lines 



( VL' + (e'h)-^ + L)e - (e«h.)h ^ ( VL^ + (e»h)^ - L)e + (e-h)h ^ 



^lU + {e-hf ' ' VLM^-h> 



These vectors, however, like those mentioned above, are not found to be im- 

 portant in electromagnetic theory. 

 56 cf. etjuation (85). 



