394 Mr. H. T. Flint on Integration Theorems 



Thus for any directed volume V we have : 



V = ?' 2 i 3 i 4 Vj — WxV 2 + ?V'i?2 V 3 — i r i 2 i 3 V 4 . . (3 • 4) 



The volume element dr may be divided into elementary 

 components. So that : 



dr — i 2 i 3 t 4 ^ f??/ <^ — ?' 3 i 4 z -, dy du dx -4- i 4 ?i?9 du dx dy — i\i 2 i% dx dydz. 



' . . . (3-5) 



The remaining part of the complete product p\p 2 pz is 

 Y1P1P2P3, an d on expansion it is seen to be : 



— i 1 a 1 X«2 a 3 + h{bi {^2°%) + c i ( a 2 c ?>) + di(a 2 d-z) } + similar terms. 



... (3-6) 



The coefficient of the second ^ is the OX component of 

 the vector product of a four- and six-vector employed in 

 Minkowski's Calculus. It may be represented in our 

 notation as : 



VifaV***) (3-7) 



§ 4. The product of four vectors will consist of terms in 

 hhhh? others in i 2 i 3 which have arisen from such products 

 as i 2 ij, 3 i^ and so on, and finally there will be terms inde- 

 pendent of z's arising from such products as i 2 i%hh which 

 reduce to 



h l dh l 3 =z — HHHH — — !• 

 Thus 



PlP2P^p4, = ^ T 4Plp2p3pi + V 2 plp 2 pZpi + Vo/OiW3f>4- ^'^ 



On expansion we find : 



^4PiP2P3p4 = hhhh( a i h 2^d i ). . . . (4-2) 



Thus the tensor of V 4 /Oi/o 2 ^3/o 4 , or TY^p 1 p 2 p 3 p 4: as it will be 

 written, is the volume of the four-dimensional parallelepiped 

 bounded by the four vectors. 



It is not possible to write iii 2 hh— ±1 by analogy with the 

 corresponding ijk= — 1, for this would lead to incon- 

 sistencies. 



Suppose we write ij 2 i%i 4 = —I. 



Then 



and 



hhhhh=—h 

 HhHHH — — H' 



The first gives: i 2 hh 



■u 



the second 



nw , = 



