340 



SCIENCE 



[N. S. Vol. L. No. 1293 



Only wlien the number of these unknown 

 quantities is reduced to one, does it become 

 possible for them to make use of the mathe- 

 matical reasoning, involving the solution of 

 functional equations, -which has led in the 

 hands of Tolman to such a wide variety of 

 useful results. Since x is the ratio between 

 their standards of length-measurement, and y 

 that between their standards of time-measure- 

 ment, they can express y in tei-ms of x if they 

 agree to make the same numerical report 

 about any one kind of quantity which involves 

 both length and time for its definition, that is, 

 if they agree to make the same reports either 

 about velocities or about accelerations. By 

 making a corresponding second agreement 

 about some quantity which involves for its 

 definition force and time or length or hath, 

 such for instance as charge or mass or energy, 

 they will be able to express s in terms of x — 

 and to derive an entire set of transformation 

 equations which involve only one unknown 

 quantity. "With this new set of equations at 

 hand, they may imdertake to set up functional 

 expressions and to derive laws as Tolman has 

 done. 



Evidently the number of the different pos- 

 sible sets of transformation equations is quite 

 considerable, for there are many measurable 

 quantities in physics which involve for their 

 definition more than one of the three funda- 

 mental undefined quantities. I have calcu- 

 lated nine such different sets. Several of 

 them lead to some of the conclusions which 

 may be deduced from the equations of and 

 0' above (based upon agreements concerning 

 velocities and charges) ; several of them lead, 

 in cases where the set of and 0' has proved 

 fertile, to insoluble or absurd functional 

 equations which point to no solution. Some 

 of them lead to laws which are contrary to 

 those whose validity has been established by 

 experiment. None of the sets is as fertile or 

 leads to as many well established laws as the 

 set which is based upon the agreement of 

 and 0' to report all charges and all velocities 

 by the same numerical value. But this agree- 

 ment is the only thing in the way of an 

 assumpUon which is involved in the simpli- 



fied form of Tolman's principle of similitude 

 that is developed by and 0' of this paper. 

 The note-worthy success of Tolman in deriving 

 from his principle a large number of experi- 

 mentally valid laws is evidence that an 

 agreement between observers working with 

 different standards of measurement to report 

 the same charges and velocities by the same 

 number is somehow more intimately in har- 

 mony with the order of nature than any other 

 similar agreement relative to some other^ of 

 the quantities of physics. 



Electrical charges may be regarded as if 

 they are made up of a countable number of 

 small units. This has been adequately demon- 

 strated by the researches of Millikan and 

 others in which electrons have actually been 

 isolated and counted. But it could also have 

 been predicted — for, as an assumption, it leads 

 in the hands of and 0' to many conclusions 

 which are otherwise verified by experimental 

 fact. In the same way the assumption that 

 velocities are of such sort that there is only 

 one right way to report their magnitude, is 

 one which leads, vastly better than any similar 

 assumption, to the deduction of laws which 

 are established in fact. Hence the assump- 

 tion is probably true. 



Professor Tolman has kindly read the first 

 draft of this paper. He suggests that the con- 

 clusion that velocity is of such nature that 

 there is only one right way to report its mag- 

 nitude, a conclusion which has here been 

 reached by abstract reasoning, may be inter- 

 preted concretely to mean that " any given 

 velocity is most sensibly regarded as a given 

 fraction of the maximum possible velocity, 

 namely that of light." 



Tennet L. Davis 



Department of Chemistry, 

 Massachusetts Institute or Technoloqt, 

 Cambridge, Mass. 



JACQUES DANNE 



With the outbreak of the world conflict in 

 1914 Le Radium set once ceased publication, 

 all of its editors being called into service. 

 The decision for service was no less definite 

 than the assurance that publication would be 



