402 Mr. H. T. Flint on Integration Theorems 



We may also define Minkowski's lor P 2 from (11*3), this 

 expression denoting the Lorentz operator, equivalent to the 

 vector divergence. 



LorP 2 is defined by the equation : 



+ jj V {A, 1 P 3 ) = - JX) Vo (lor P 2 . Ur) 



| 

 = -JJ"JV„{WtD 1 P 2 }, 



since the part V±drDi, which with V 2 drD 1 makes up the 

 product drT>i. cannot add to the scalar product. 

 Now Idr is a product of the type V 1 so that 



Y (I^T.D 1 .P 2 ) = V (IrfT.V 1 D 1 P 2 ) J 

 lorP 2 =V 1 D 1 P 2 *. 



Since by § 4, Y (dv'P 9 )=Y (dv' P a r ), 



we have by (12*4) and (12*2) 



J j V (di/P,) = jf V (^'P S ')= - f j j V (lor P/ . I dr). (11-5) 



Thus in integrating the normal components of P 2 over a 

 surface, the operation lorP 2 occurs, while for tangential 

 components (i. e. components of P 2 along dv) lor P 2 is 

 replaced by lorP 2 '. 



This is important in the electromagnetic equations. 



§ 12. The held equations of the electron theory as expressed 

 by Minkowski may now be regarded as a generalization of 

 Gauss's integral of intensity over a surface. 



Putting these integrals in their usual form we have : 



jD n 6?S=— \pdv or ^SDdv = ^pdv, 

 jfB n rfS = or JSB^ = 0. 



If now we form two six-vectors : 

 P 2 = ( — iD x ) i 2 i 3 + ( - iDy)^ + ( — i Y* z )i A L + EUV 4 + H^ 4 



+ Hy 3 ; 4 , 



Q 2 = iBJ 2 i s + +E x i^+ 



where i= \/ — 1. D x etc. denote the components of displace- 

 ment as usual, and H, B, and E have their usual significance. 



* Cf. Cunningham, 'Theory of Relativity,' pp. 101-2. 



