LEWIS. — ON FOUR-DIMENSIONAL VECTOR ANALYSIS. 175 



The complement of a 2 -vector is the completely perpendicular 2 -vector. 

 In the vector analysis at present in use it is customary to identify a 

 vector with its complement, and this is also done by Abraham ^^ in 

 the paper in which he makes use of four-dimensional vector analysis. 

 In our present analysis there is no advantage to be gained by this step, 

 which may cause much confusion. 



As in the case of three dimensions we may define a differential 

 operator, having the form of a 1-vector, as follows : 



0= ki — + k.3 — + kg — + k4 — . (46) 



. da\ d.r.2 0^3 du\ 



This operator ^^ (read " quad ") may be treated like a simple vector 

 under the same conditions as in the case of V . We thus obtain a num- 

 ber of important equations such as the following. 



0* = k,«^ + k.^ + k3|i + k4-*. (47) 



diCi 0X2 0^3 oXi 



0^ = p: + p + ?^ + lh, (48) 



ou'i oj'o oj's da^i 



^-=G^:-l:)>^'=+(i:-aT:)>=.'+--- («> 



These three expressions correspond to gradient, divergence and curl in 

 three-dimensional analysis. We may also apply O to vectors of higher 

 orders, for example, by (43) and (38), 



O A= f ^ + ^-^ + ^-^] K + (^-^ + -^^' + ^-^^ k. -f . . . 



\ 0^2 dxz dXi I ^ \ d.Vi dxs da't / " 



(50) 



We may form other operators like, AO, a 1-vector operator, and the 

 scalar operators a O, and </^. The last is the very important operator 

 which Lorentz in a special case calls the d'Alembertian operator, 



g2 q2 q2 q2 



^* Abraham (loc. cit.). 



^■^ The operator O has the same scalar components as the operator lor used 

 by Minkowski. 



