658 



SCIENCE 



[N. S. Vol. XXIX. No. 747 



mass, imparts momentum dM in time dt and 

 so, 



f = dM/dt. (2) 



With (1) and (2), 



dE/dM = V. (3) 



But a momentum from a mass dm moving 

 with a velocity V requires that 



Vdm = dM, 



and so with (3), 



dm = dE/Y' 



(4) 

 (5) 



Now equation (5) is a very simple thing. It 

 gives the mass needed at velocity T to pro- 

 duce the energy dE in this special way. But 

 Professor Lewis says this equation gives the 

 change in mass when the energy of the body 

 changes hy dE in any manner whatsoever. I 

 do not see that this inference is legitimate at 

 all. 



In the fourth paragraph there is the start- 

 ling statement that the mass of a moving 

 body becomes infinite at the velocity of light. 

 It seems to me this at once throws suspicion 

 on the line of reasoning leading up to such a 

 conclusion. Professor Lewis recognizes this 

 difficulty for he says, " Therefore that which 

 in a beam of light has mass, momentum and 

 energy, and is traveling with the velocity of 

 light, would have no energy, momentum or 

 mass if it were at rest, or, indeed, if it were 

 moving with a velocity even by the smallest 

 fraction less than that of light," adding with 

 great naivete, " After this extraordinary con- 

 clusion it would at present be idle to discuss 

 whether the same substance or thing which 

 carries the radiation from the emitting body 

 continues to carry it through space, or, indeed, 

 whether there is any substance or thing con- 

 nected with the process." (Italics mine, C. L. 

 S.) Moreover, I do not see how this part of 

 the fourth paragraph is consistent with (5). 

 There is no special value, numerically, to be 

 assigned to V in deducing (5) and so there can 

 not be an extraordinary jump from a finite to 

 an infinite value when V has a certain finite 

 value assigned it. We have no right to as- 

 sume that the velocity of light is the greatest 



possible velocity in the universe. What would 

 the mass become for a greater velocity? What 

 does the mass become for the lesser velocity 

 of light in water? 



It seems to me that there is no need for any 

 such startling conclusion. In fact, no oppor- 

 tunity for it, as I think will be seen from the 

 following. 



A beam of radiant energy composed of a 

 moving mass changes the momentum of the 

 body struck by it both by the change in 

 velocity and by the change in mass due to the 

 mass of the beam passing into the body struck 

 by the beam. Hence, from the definition of 

 momentum, M = mv, 



dM = md/D ■\- vdm. ( 6 ) 



Eeplacing in (4), 



Vdm = mdv -{- vdm, 

 or 



dm/m = dv/ {V — v). 



In this equation, V is the velocity of the 

 striking mass of the beam, the mass of which 

 is dm, while v is the velocity of the object 

 struck whose mass is m, and dv is the velocity 

 imparted to it, to the mass m. Consequently, 

 this equation expresses the relation between 

 the change in mass of the object struck, due to 

 the accretion from the mass of the beam, and 

 V, the velocity of the object, due to the impact 

 of the beam mass. Integrating, 



TO/m'^y/CF — v) 



(7) 



where m° is the mass of the object at rest, 

 that is, when v is zero. When v is F, its mass 

 becomes infinite, which means that a mass 

 aggregating to an infinite mass must accumu- 

 late on the object before it will attain a 

 velocity equal to the velocity of the pelting 

 mass of the beam. In other words, the mass 

 of the object must become relatively zero and 

 not absorb any of the kinetic energy of tha 

 beam for itself, to increase its motion. This 

 is surely simple! Who would conclude from 

 this that when a body is given a velocity equal 

 that of light in any way whatsoever its mass 

 becomes infinite? Yet this is what Professor 

 Lewis seems to do. He deduces his equation 

 in a somewhat different way, passing through 

 the energy and not through the momentum. 



