382 Dr. L. Silberstein on a Time-Scale 



deflexion produced, we Lave P.2A.r=X0. Also if the 

 period of the suspended system be t, then £ = 27T\/I/A, where 

 I is the moment of inertia. Eliminating \ gives 

 F = 2-rr 2 Id/rAt 2 . When D is the scale deflexion of a spot of 

 light on a scale at distance d 



2tt 2 ID 

 r_ rAt'd [) 



Combining (1) and (2) gives Knudsen's expression 



4tt 2 ID T 



P ~ rAt 2 dT,-T' 



Royal Grammar School, 

 Newcastle-on-Tyne. 



XXXIV. A Time-Scale independent oj Space Measurement. 

 Parabolic and Hyperbolic Kinematics. By L. SlLBERSTElK, 

 Ph.D., Lecturer in Mathematical Physics at the University 

 of Rome *. 



1. PT^HE principle of common, metrical time-scales amounts 

 L to this: — A certain kind of, say, rectilinear motion 

 of a particle f is dechired to be a uniform motion, which 

 logically plays the part of an undefined term. The path of 

 the particle is cut up, by means of compasses or other rigid 

 transferers, into a series of equal segments, and the instants 

 oE passage of the mobile through the divisions of this metrical 

 scale of lengths are taken as t = 0, t— 1, 2, 3, and so on. The 

 instants and 1 being fixed arbitrarily, all other instants are 

 constructed with the aid of a rigid length (or angle) transferer. 

 The history of the moving particle (or of a light flash) is thus 

 dissected by an extraneous process, time is subdivided by 

 appealing to measurement of space, which process, moreover, 

 implies the concept of rigid bodies. 



Without questioning the high practical advantages offered 

 by this, the usual and only known, chronometrical procedure, 

 1 propose to show how a time-scale can be set up without the 

 aid of space measurement, in fact, without any space scale of 

 points whatever. For reasons which will appear hereafter 

 I propose to call such an emancipated scale of instants 

 a projective time-scale. Any three distinct instants, say 

 marked by three bell-strokes, being arbitrarily chosen, an 



* Communicated by the Author. 



f In practical chronometry a motion of rotation round an axis is used. 

 This, however, does not change the nature of the argument. 



