WILSON AM) LEWIS. — RELATIVITY. 407 



notable when taken in connection with the recent theories reganUng 

 the constitution of Hght, embodied in the quantum hypothesis. 



Let us for simplicity first consider such cases as arise in our two 

 dimensional j;;eomctry. Consider a material rod of infinitesimal cross 

 section moviiiff uniforndy in its own direction. Suppose now that 

 we regard this rod as made up of discrete particles. Then in our 

 geometrical representation each particle will give rise to a vector 

 of extended momentum ///oW, and these vectors will all l)e parallel. 

 The whole space-time locus of the rod will l)e a set of parallel (5)-lines. 

 The rod as a spacial object possessing length has no meaning until a 

 definite set of space-time axes have been chosen, and this choice is 

 arbitrary. There is, however, one such choice which is unique, and 

 that is the selection of the time-axis along w, and the space-axis per- 

 pendicular thereto. In this system the mass of each particle is its 

 /»o. 'iiicl the sum of the m^'s of any segment of the rod divided by the 

 length of the segment is the average density. If the particles are 

 sufficiently numerous, we may regard the rod as continuous and re- 

 place conceptually the locus of the rod as a set of discrete (5)-lines 

 by a vector field continuous between the two (5)-lines which mark 

 the termini of the rod, and represented at each point by a vector 

 parallel to w and equal in magnitude to the density at that point. 

 This is the density as it appears to an observer at rest with respect to 

 the rod, and may be called mq. The vector /zqW has therefore a defi- 

 nite four dimensional significance. Its projections on any arbitrarily 

 chosen space and time axes are, however, not respectively the density 

 of momentum and mass in that system. For 



MoW= -.^°=^(v+k4). (87) 



But n, the density in this system, is not equal to mo/ "^1 — «^ but 



, = ~^, (88) 



as the units of mass and length both change with a change of axes. 

 Conversely we may replace a continuous by a discrete distribution. 

 Let us consider a continuous vector field f of (5)-lines. Then any 

 region of this field, embraced between two (5)-lines sufficiently near 

 together, may be replaced by one or several parallel (6)-vectors, of 

 which the sum is ecjual to f multiplied by the length of the line drawn 

 between and perpendicular to the boundary (6)-lines. We may also 



