398 Mr. H. T. Flint on Integration Theorems 



This mode of arriving at D gives to it the inconvenient 

 canonical form : 



• • • d . . . B • . • O ...."& fC.Q\ 



§ 7. The circuital theorems of ordinary vector analysis 

 SdpT(p)=flY(dvV).¥(p) 



and ^ dvF(p) = JJ J V F(/o) . ^ 



are generalizations of Stokes's and Gauss's theorems. We 

 proceed to establish the corresponding circuital theorems for 

 four-vectors. They are three in number and the operator V 

 is now replaced by ~D lm 



§8. In order to establish the theorem for integration along 

 a line we require the relation : 



<*iV a 2 a — a 2 V a 1 « = V 1 (V 2 a 1 a 2 • *)• • • (8*1) 

 If we write : a Y = i x ai + ij)^ -f hc 1 + i 4 d l5 

 a 2 = i Y a 2 + z 2 6 2 + i B c 2 + t 4 <^ 

 a =«\a +i- 2 o +2 3 c -fi 4 ^, 

 we find that the coefficient of i x on the left is 



a 2 [acLi + bbi + cc l + ddi) — a x (aa 2 + bb 2 + cc 2 + dd 2 ) , 

 while on the right the coefficient is 



— b{a l b 2 ) +c(g'!«2) —d(a 1 d 2 ), 

 and these are of course identical. 



§ 9. Let M be at the extremity of the vector p and 



Fig. 1. 



D 



construct about it as centre a small parallelogram bounded 

 by the vectors dpi and dp 2 as in the figure. 



