OCTOBES 10, 1919] 



SCIENCE 



339 



measure all quantities whicli come witliiii his 

 observation. The standards of the two ob- 

 servers, and 0', may be connected by al- 

 gebraic equations, for length, l' = x I, for time 

 i' = ytj and for force, /' = zf, where x, y and z 

 are constants which may have any value and 

 which express the ratio between the magni- 

 tudes of the corresponding standards of the two 

 observers. From these three equations corre- 

 sponding " transformation equations," involv- 

 ing X, y and z, for all of the measureable quan- 

 tities of physics may be calculated. Suppose 

 now that the two observers got together and 

 says to 0' : 



Surely it is the privilege of each of us to work 

 with whatever standards he prefers, you prefer 

 foot-seconds while I prefer gram-centimeters; yet 

 let us, for such and such reasons and for our mutual 

 convenience, each still keeping his standards dif- 

 ferent from those of the other, so alter our stand- 

 ards that we can both report all electrical charges 

 and all velocities by precisely the same numerical 

 value. 



0' may be conceived to reply: 



We know indeed that electrical charges are made 

 up of a number of ultimate charges or electrons 

 and that each of these little charges has the same 

 magnitude. If we report all charges by the same 

 number, that is virtually the same as each of us 

 reporting the same count. I see that in favor of 



cause it seems to me to be primary in the course of 

 experience, not because it makes any difference in 

 the general argument. Many physicists, perhaps 

 most, are disposed to believe that there are two 

 additional fundamental and ultimately undefined 

 quantities necessary to complete the structure of 

 modern physics, namely, (1) temperature or en- 

 tropy (for thermodynamics) and (2) quantity of 

 electricity or magnetic permeability (for electro- 

 magnetics). But temperature is defined by the 

 ' gas-law, and charge is defined by Coulomb's law. 

 These laws, discovered by the experimentation of 

 workers in whose minds temperatures and charges 

 (respectively) were regarded as equal if in similar 

 situations they produced equal effects, have gradu- 

 ally come in the history of science to assume the 

 sacred functions reserved for definitions. It can 

 not be asserted too strongly that they are not 

 "laws." This, however, is methodology, not theo- 

 retical physics, and not the central argument of 

 the present paper. 



your suggestion, but I know of no corresponding 

 consideration in regard to velocity. I see that 

 there is only one really correct way of reporting 

 the magnitude of charges. But I do not see that 

 such is the case for velocities. 



Suppose however, that and 0' do put 

 the agreement into eifect. Then v' = v, and 

 e'^ e. 

 Then 



l/ = — = — = iv) = 

 (/ yt y^ ' 



and X ■■ 



Also 



x^--<jfP = x-^-e = i'; and z=-2- 



Transformation equations involving only one 

 unknown quantity, x, may now be written con- 

 necting all of the measurable quantities of 

 with the corresponding quantities of 0'. 

 These equations are identical with those of 

 Tolman ; and and 0', working with the per- 

 fectly legitimate^ mathematical reasoning that 

 Tolman uses, may derive, by virtue of their 

 simple agreement, all of the conclusions which 

 he derives from his principle of similitude. 



So long as and 0' have no agreement 

 between themselves, the transformation equa- 

 tions connecting their various quantities will 

 involve three unknown quantities, x y and z. 



3 To illustrate the method of reasoning — The 

 transformation equation for energy density is 



and that for temperature, 



T' = -T. 



If energy density is a function solely of tempera- 

 ture, then u^F{T) and «' = F(T') where F in 

 both equations has precisely the same value, for 

 both observers obviously will find the same law. 

 Substituting in the equation u' = F{T'), we have 



h'-^iH 



But u:^F(I), hence 



li^(r) = i^(^r). 



The only solution of this functional equation is 

 F(a.)=fco*. Hence u=^hT^, and Stefan's law 

 has been demonstrated without experiment. 



