WILSON AND LEWIS. — REI^TIVITY. 465 



Hence \}C hrinfjs in, ;is iniffht be expected, tlie rate of change of 

 curvature, just as 0W brought in the curvature. We have 



= n(- W + ' \'"l) IxC - I (l + I lw)xc + jI (- I ;;^^)xl 

 + „,(1+1.C) -W+ J, 1 lxW-^3 C-., , .1 Ixw 



R*' ' ' \ ' R J R' V li (Is 



^3 ^1+^^lWjxW ^3 ^Cxl. 



In this expression the product indicated by the cross is always per- 

 formed first, regardless of the parentheses. If now the cross be 

 inserted to find O^O^P* the result O^O^P = is obtained, as 

 required by equation (51). Moreover, if the dot be inserted so as to 

 find <3> • (O^P). the result is also 



O-Oxp = 0. (85) 



We have, of course, proved this theorem only for points lying off the 

 given (5)-curve. 



We have the mathematical relation (55), namely, 



0-Oxp = 0«>-P)-(C>-<C>)p. 



But we have seen that <C>*P = 0, and therefore 



O-Op=O'P=0. (86) 



The existence of this extended Laplacian equation justifies the use 

 of the term potential *° for p. 



40 It is interesting to enquire what form the potential p might be given other 

 than VT/R. Suppose that p should be independent of the curvature of the («)- 

 curve. The only vectors then entering into the determination of p at any 

 point Q would he w and 1. The only possible form of a l-vector potential 

 would therefore be 



p = ^(/.-jw +/(/i;)i, 



where R = — l«w. The expression for <^p becomes 



Op = /(«)(- w + ^ V'^'O'' " ^'^4^° 



+ /' (72) - w + 



1 + l»c 

 R 



\Y+f{R)[l + )^lw). 



