WILSON AND LEWIS. — RELATIVITY. 491 



If we choose a particular k^ axis and its perpeiidiciilar plaiioid, then 

 dQ* = (/5 kj and the above expression becomes 



M(k.,.M).M+ (k4-M*)-M*]J3. (146) 



We may perform the operations here indicated upon the expanded 

 form (140) of M and obtain^ ^ 



[exh+He'+ /'')k4]r/S. (147) 



Now exh, the complement in three dimensional space of e^h, and 

 ^(e^ + h') are the familiar expressions for the Poynting vector and the 

 density of electromagnetic energy, and the above expression therefore 

 represents what is ordinarily regarded as the total electromagnetic 

 momentum and energy in the \olume dB. 



Now after the axes have been chosen we may perform similar 

 operations with ki, ki, ks. Thus 



i (ki.M)-M + i (k,.M*).M* = A%ki+ A>o + A'^ks - X(k4, 

 i(ko.M).M + i(k2-M*).M* = }'xki + l',k, + l\ks — Ytku 

 i(k3-M).M + i(k3-M*).M* = Z,ki + Zyk; + Z.ks - Ztki, 



i(k4.M).M+ i(k4'M*).]V[* = 7',k, + ^ko + T.ks + Ttki, 



where 



X. = i (e,' - e-^ - e^' + h' - /^,- - h'), 



Yy = I (e-^ - t'3- - e{' + }i2' - h{- - h^), 

 Zz=\ {e,^ - e^~ - ei + U - K^ - hi), 

 Tt = h (cr + ei + ei + h' + hi + hi), 

 Xy = Yx = 6162 + hif^i, etc., 

 Tx = Xt = 62^3 — e-ih, etc. 



In these equations X^, etc., are the familiar expressions for the 

 components of the Max^vell strains; Tx, Ty, T^ are the components 

 of the Poynting vector; and T^ is that which is ordinarily assumed 

 to be the density of electromagnetic energy. This procedure is 

 essentially that of Minkowski. We may reproduce his procedure 

 exactly with the aid of dyadics. It may readily be shown (see appen- 

 dix, § 62) that if M is any 2-vector, and I the unit dyadic or idemfactor, 

 then the dyadics 



$ = (I.M).(I-M) <J>* = (I.M*).(I.M*) 



57 For abbreviated methods see a footnote in § 53. 



