electrodynamics in the Maxwell and in the Einstein forms 111 



Thus the matrix ]}' is the inverse of jo; so that if P^g be the co- 

 factor of jj^g in the determinant, td, of ]}, we have jpj = Pj-s/^' 

 With this transformation the quadric differential form 



{abcduvwfgh^dx, dy, dz, dt)^ 



changes into another such form which is written 



{a', ... ,h'\dx',dy',dz',dt'f. 



Using A' for the matrix of this latter, we thus have 



A'= pl^j), A = p'A'jo', 



where, as usual, p denotes the matrix obtained from p by inter- 

 change of rows and columns. 



Corresponding to this transformation, we introduce new func- 

 tions Mij' .... These may be defined by the fact that if 



{dx, dy, dz, dt) = jp {dx', dy', dz', dt'), 



{hx, Sy, 8z, 8t) = p {8x', 8y', 8z', 8t'), 

 then 



S {M^ {dySz - dz8y) + if 14 {dx8t - dt8x)} 



= S {M23' {dy'8z'- dz'8y') + M^^ {dx'8t' - dt'8x')}. 



Thus, in matrix notation 



— m {dx, dy, dz, dt\8x, 8y, 8z, 8t) 



= -m' {dx', dy', dz', dt') {8x' , 8y', 8z' , 8t'), 



leading to m'= pmp, m = p'm'p'. 



If, as before, the determinants of p, m, A be m, jx^, — e^, and 

 the determinants of m' , A' be similarly fx"^, — e'^, these equations, 

 with A'= j)Aj9, give e'^ = ta^e^, /u,'^ = ta^ju,^ and hence, with a 



proper sign for e', which is (— 8')^, we have e l(x' = e//x. 



The matrix n is replaced after the transformation by a matrix n' 

 connected with m' as was n with m, namely by 



eV= /x'A' (m')-W 



= fx'pAp . p~^m~'^p~^ . pAp 



= fx'pAm-^Ap 



SO that n'^ pup, n = p'n'p' . 



The matrix n is thus transformed by the same rule as was m. 



We now introduce Grassman's units e^, 62; 63, 64, obeying the 

 laws 



VOL. XX. PART I. 12 



