492 



PROCEEDINGS OF THE AMERICAN ACADEMY, 



are such that 



a.$ = (8i'M)'M 



a-4>* = (a-M*).M*, 



where a is any 1 -vector. The expressions which we obtained from IVT 

 and ki, k2, . . . in the form 



i(ki-M).M+ §(krM*).lV[*, etc. 



might therefore equally well have been written 



ik,.('J> + ^*), etc. 



It is these latter expressions which Minkowski obtained. The dyadic 

 I ($ + $*) is identical with Minkowski's matrix S, except in as far 

 as he used imaginary space, and distinguished between electric force 

 and displacement and between magnetic force and induction. ^^ 



While, as we see, the use of the dyadic J(^ + ^*) yields no results 

 which are not also obtainable by the methods of simple vector analysis, 

 yet to one who is familiar with the dyadic method it frequently affords 

 a considerable gain in simplicity. Thus for example we may obtain 

 an important result by considering the expression i<^* ($ + <!>*), 

 which may be shown to vanish in free space. ^^ Now, if ^s be the 

 three dimensional dyadic of the Maxwell strains, if exh is the Poynting 

 vector, and if Tt is the density of energy, we have 



= lO- (* + •^*) = O- (^. — exhk4 - k4exh — k4k4rt), (148) 

 or 



V-^. — Y. (exh) = and V-exh + ^^Tt= 0. (149) 

 at dt 



The first is the important equation of Lorentz connecting the force 



58 The form of the dyadic -^ = ^ (^ + **) is 



Zxkiki + Xykik.. + A'.kika - X(kik4 

 + F^k.ki + Fj/k.k. + y^k.ks - Fik2k4 

 + Zxkaki + Zj/k3ko + Z^ksks - Z,k3k4 

 - 7':rk4ki - Tyia^-kn - T^k,kz - Ttk^a,. 



59 From (158), with A = M, A' = M, and from (141) and (142), since in 

 free space q = 0. Where there is electricity the equation would be 



i<^.^ = 47rq-M. 



