LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 177 



Following Minkowski we may define a 1- vector, q, in the space x, y, z, 

 ict (or Xi, X2, Xs, Xi) of which the projection on the 3-space is -v and 

 the scalar component along the a'4 (or ict) axis is Iq, by the equation, 



q = ^Y + i(jki, (59) 



or q = - Viki + - ^2^2 + - t'skg + / q k^. (60) 



c c c 



Furthermore, from the electrical force e, and the magnetic force h, we 

 shall find it convenient to define two new 2-vectors, E and H, by the 

 equations, ^^ 



E = — /exk4, (61) 



and H = hkio,. (62) 



H is the 2-vector complementary to h in the 3-space x, y, z. From 

 these definitions we have 



Hu = h ; //23 = h ; //31 = K (63) 



-£'14 = — «<^i ; ^24 = — ^'^2 ; -£'34 = — «^3- (64) 



From H and e we may define "^^ the vector potential a and the scalar 

 potential by the familiar equations, 



H = V xa, (65) 



-e=v* + i^. (66) 



Finally we shall define a new 1 -vector m by the equation, 



m = a -f /(/.k,, (67) 



or m = «iki + a^i + ^s^s + i<i^i. (68) 



Thus m is a vector of which the projection on the 3-space .r, y, z is the 

 vector potential, and of which the scalar component in the ict direction 

 is the scalar potential multiplied by V— 1- 



" Compare in this connection the discussion of e as a "polar" vector, h as 

 an "axial" vector, in Abraham-Foppl (1, p. 243). 



^° This definition is evidently not complete, since a and </> are derived from 

 H and e by a process of integration. We shall return to this point. 

 VOL. XLVI. — 12 



