of Four-Dimensional Vector Analysis. 397 



it is convenient to adopt a rule of signs for the three-dimen- 

 sional volumes that may be chosen from them. The rule 

 adopted is to define the volumes as : 



+ ^3p2p3Pi, —^Bp3p4pl, +^ T 3p4Plp2, —^3PlP2pd- 



This is to be associated with the rule of § 3. The volumes 

 corresponding to the vectors hdx, i 2 dy, i%dz, i A du are : 



i 2 i 3 z 4 dy dzdu, — i^hh dz du dx, ifaiz du dx dy, 



an d — iii 2 h dx dy dz. 



§ 6. A vector operator D x defined as 



plays an important part in four-vector analysis just as the 

 Hamiltonian V does in three dimensions. 



The suffix D x serves to show that it is of the type of a 

 vector Vp 



Any function a of p will undergo a small change da, on 

 account of a variation d(j> given by the equation 



da=-Y dpD 1 ,a (6-1) 



(6*1) is the same result as is expressed in Cartesians by : 



da==^- dx+ ^- dy + ~~-dz + ^— . du. 

 Ox l$y J l$z ^u 



This method of introducing D x appears to be the simplest. 

 V is sometimes defined by the operation : 



Lin Jcfo.F(p) 



r_»0 7~. = VF( »> 



where dv denotes a divided element of area in two dimensions 

 bounding a small volume v, and F is any function of p, the 

 vector from the origin to dv. 



If this method of defining an operator be extended to four 

 dimensions we shall have : 



l-L«w =])FW> . . . (6 . 2) 



where dristhe directed element of three-dimensional volume 

 and v a small four-dimensional volume. 



