electrodynamics in the Maxwell and in the Einstein forms 179 



Next let ^, t], ..., I', Tj' , ... denote respectively 



d^ ^ d d 



dx' dy' "•' dx"' dy""" 

 We then have 



(f , 7?', r, t') = p (f , r^, I r). 



Hence • e'^', = e{ |'+ e2'->?'+ ^3'^'+ e/r', 



= |'e'= j;| . e'= pe'i = e|. 



Now consider the differential equations. They are capable of 

 very succinct expression. Put 



Ml = - 7]Ms^ + CM2, - TM23, M2 = - ^ifi4 + ^1/34 - TM31, 



Mg = - 1^24 + 77M14 - tMi2, ilf4 = ^^23 + V^si + C^l^ia, 



these being the forms occurring on the left in these equations 

 (p. 174). Denote the row of these four quantities by 



m(i) = (Ml, M2, M3, M4). 



The expression e^ . me^, whose meaning is 



- {e^i + e^T) + gg^ + e^r) 



(62^3^23 + egCiMai + 6162^12 + eie4Mi4 + 6264^^4 + 6364^^34) > 



is at once found on evaluation to be equal to 



^iMi + E^M^ + ^3^3 + E^M^, 



which we can represent by Em^^\ In other words we have 



ei . me^ = Em^^K 



Thus the original differential equations are 



ei . me^ = p {E^V, + E^V, + E^V , + ^4)) .tt. 



e^.ne^ = j ^ '' 



In this form it is easy to show that they are unaltered by the 

 transformation, provided suitable values be adopted for the new 

 values of p, V^, Vy, V^. 



For we have shown that the operators ef , e'^' are equal, and 

 that me^, ne^ are equal respectively to m'e"^, n'e"^. Also that 



a) 



Thus, as e'^' . m'e"^ is identically equal to 



^i'Mi'+ ... + ^4'M4', 



that is to -p {E) {M') = - E . pM', 



we have {wM^, wM^, wM^, wM^) = p (Mi', M2', M3', M/), 



12—2 



