488 PROCEEDINGS OF THE AMERICAN ACADEMY. 



locus of the electron. In the case of a continuous distribution of 

 electricity we have ^* 



O'm = - 4xq, (137) 



which might be proved directly; but this is unnecessary since it has 

 frequently been shown by familiar methods that 



O'a = — 47rpv and O'0 = — 4xp. (138) 



Furthermore it is unnecessary to evaluate once more in detail the 

 2-vector 



M = Oxm = Vxa + (v0 + ^W (139) 



For Vxa is the three dimensional complement of Avhat is ordinarily 

 known as curl a or h, and V0 + a = — e. Hence 



M = H + E, 



where the components of H and E are once more the components of 

 magnetic and electric force. 



56. Whether the 2-vector M of extended electric and magnetic 

 force be derived from a number of point charges or from a charge 

 continuously distributed, it is in general a complex or biplanar 2-vec- 

 tor.^^ The two invariants of M are M-M and M«M* = (MxM)*. 

 If, after choosing space and time axes, we write 



M = hik23 + /?2k,3i + hki2 — fiku — e2k24 — e^ksi, , . 



m* = C,k,, + f2k31 + Cskr, + h^ku + /^2k24 + A3k34, ^ ^ 



54 The vector 4 ttQ which we use is identical with the vector q used by Lewis, 

 owing to a different choice of units of electrical quantity. 



55 Since it is customary to divide a complex 2-vector into the two complcteh' 

 perpendicular uniplanar vectors which are uniquely determined, one being a 

 (7)-vector, the other a (5)-vector, we might expect that the two hnes of inter- 

 section of the (6)-plane with the hypercone, and their projections upon a 

 chosen space, might prove important. This is, however, not the case, although 

 indeed from an analytic point of view the four directions, two of them imagi- 

 nary, in which the hypercone is cut by the completely perpendicular (s)- 

 vector and (7)-vector form a set of four independent directions possessing 

 some advantages over the system ki, k?, ks, k4. In fact four vectors ji, jj, 

 j», J4 can be selected along these directions such that 



jl*jl=J2*J2=J3*J3=J4*J4 = 0, jl*J2=J3*J4=l, 



jl*J3=jl*J4=J2'J3=J2*J4=0. 



In terms of such a set of vectors the differential of arc is given by the equation 



dT-dr = dx^ + d}f + dz^ - df = Adudv + Bdwds. 



(See Bateman, Proc. Lond. Math. Soc. [2] 10, 107). 



Other vectors which might be thought important would be the two lines- 

 in which the completely perpendicular planes cut the planoid which is taken. 



