WILSON AND LEWIS. — UELATIVITY. 423 



problems in Newtonian mechanics proves equally applicable in the 

 new mechanics. 



If the impact, instead of being perfectly elastic, were such that the 

 particles remained together after the collision, the two vectors a and b 

 would merely be merged into a single vector a + b. The sum of the 

 wo's would not in this case remain constant, but would be increased 

 by the heat (or mass) produced by the impact and obtained from the 

 "kinetic energy" of the relative motion. This is all equivalent to 

 the simple geometrical theorem that the (5)-diagonal of a parallelo- 

 gram whose sides are (5)-lines is greater than the sum of the two 

 sides. 



24. The concepts of momentum and energy (mass) are ordinarily 

 extended from the primitive mechanical phenomena to those involving 

 so-called radiant energy. We shall see that the ascription of mass 

 and momentum to light or other radiation is in consonance with the 

 geometrical representation which we have adopted. 



Let us consider a ray of light emitted in a single line for a definite 

 interval of time. Such a ray alone can be considered in our two di- 

 mensional system. If the interval of time is very short, so that the 

 front and the rear of the ray are very near together, we may regard 

 the ray as a particle of light. The motion of such a light-particle 

 can only be represented in our geometry by a singular vector, and to 

 any observer its velocity is unity. Although the interval of any 

 singular vector is zero as compared with the interval of any (7)- or 

 (5)-vector, intervals along a given singular vector are, as we have 

 pointed out, comparable with one another.]: I 



Supposing now that a given light-particle is represented by a definite 

 singular vector, let us see whether such a vector can be regarded as 

 an extended momentum. If so, its projection on any chosen space- 

 axis must represent momentum, and its projection on the correspond- 

 ing time-axis mass or energy. These two projections must, moreover, 

 be of equal magnitude in this case, since the velocity of light is unity. 

 It is immediately obvious that this latter condition is fulfilled, since 

 the vector is singular (§11). If a is the vector, then in terms of two 

 sets of axes 



a = wki -f mki = w'ki' + m'ki'. 



If then a represents extended momentum, m must represent the mass 

 of the light to an observer stationary with respect to the first system 

 of axes, and m' the mass as it appears to an observer stationary with 

 respect to the other system. 



