WILSON AND LEWIS. — RELATIVITY. 4G9 



one boundary to the other (Figure 25), the 2-vector c^r^l is independ- 

 ent of the way in which dr was (h'awn and the 1 -vector (dTxl)*l is 

 determined, and is in a certain sense representative of the region of 

 the field chosen. 



It may be of interest to obtain the projection of 1 and (rfr^l)*l 

 upon two sets of axes ki, k4 and k/, k^' where the angle from k4 to 

 k/ is <^ = tanh"^ v. Let the vector 1 be written as 



1= a(ki + k4) = a'(ki' + k4'). 



Now by the transformation equations (7) we have 



a' = a(coshii' — sinhi/') = a , r = a -. 



Vl _ ^,2 ^1 -}- B 



Hence the ratio of the components of 1 along the new axes to the 

 components along the old axes is Vl — «/ Vl -f- v. But (drxl)* is a 

 member independent of any system of axis. Hence the ratio for 

 (c?r>^l)* 1 is the same as that for 1. 



Now while it is impossil)Ie by any four dimensional methods 

 to redistribute the vector (drxl)*l as a continuous vector field, it is 

 alwa^'s possible after arbitrary axes of space and time have been 

 chosen to make such a distribution. Thus if between the two bound- 

 ary lines dr be taken parallel to ki and dr' parallel to k'l, then as 

 before f/rxl = c/r'xl. By taking the complement of both sides and ap- 

 plying (24), then, since 1 is its own complement, we find r/r*l = dr'-l. 

 But dr-l is equal to a(/r*ki = adr, and r/r'*l = adr'. Hence 

 dr/dr' — a' /a. Thus the density of the components of the vector 

 (c?rxl)*l in the one case is to the density of the components in the 

 other case as a^ is to a'-, equal to (1 — v)/{l + v). Thus while we 

 have seen that the energy and momentum of a light-particle (§ 24) 

 appear different in the ratio Vl — v/ '^1 -{- v to two observers, if the 

 energy and momentum are regarded as distributed their densities will 

 appear different to the two observers in the ratio (1 — »)/(l + v). 



Let us proceed at once to the discussion of similar problems arising 

 in space of four dimensions. Here also it is possible to pass at will 

 from a consideration of continuous 1 -vector fields to a consideration 

 of equivalent discontinuous distributions of 1-vectors in the case of 

 all non-singular vectors, by an extension of either of the methods 

 which we have used in two dimensional space. Thus if a region of 

 the field is cut out by a (hyper-) tube of lines parallel to the vector of 

 the field, then the original vector multiplied by the volume of inter- 



