of Four-Dimensional Vector Analysis. 399 



If the value of a function be F(p) at M then along the 

 side marked dp 2 the value will be 



F(p)-iY d Pl -D 1 .F(p) (9-1) 



in accordance with § 6 ; while along the opposite side we 

 shall have the value 



F(p) + |Y rf Pl D 1 .F( P \ .... (9-1) 



and there will be similar expressions for the other two sides. 

 Thus on making the summation %dp¥(p) and making the 

 circuit in the positive direction, we find 



l l dp¥(p) = d Pl Y dp 2 J) 1 . ¥(p)-dp 2 Y d Pl T) 1 . F(p) 

 =Y 1 (F 2 d Pl dp 2 :j) 1 ).F(p) 



= Y,(dvD l ).F(p) (9-2) 



If F(p) is regular throughout a finite surface we have 

 by adding up for each element of which the surface is 

 composed : 



j-dpF(p)=^Y 1 (dv-D 1 )¥[ P ). . . (9-3) 



In the summation all the edges of the elementary area 

 except those round the boundary contribute twice to the 

 summation with opposite signs in each case, so that finally 

 the integral is taken round the boundary. 



A particular case is obtained by choosing F(p) in the 

 form i x X + z 2 Y + i 3 Z -j- iJJ, when we obtain : 



j(Xd.r+ Ydy + Zdz + Udu) 



+(S-ti)^(i-i) 



\oi/ ow/ J \ d~ d« / 



by writing dv in the form (2*9) and taking the scalar product 

 of each side. 



dxdu 



