234 PROFESSOR TAIT ON IMPACT. 



where the numerical factors are introduced for convenience. This assumes X, above, to 

 vary as the square of the distortion measured from the datum circle, and it gives, for the 

 time of compression, in terms of the greatest distortion, a, the expression 



dz 



.-/ 



P 



\l n 2 + a a 



to a sufficient approximation. Here p is a numerical quantity which is about 1 *6 when 



a/a is small in comparison with n 2 , and continuously approaches the value 1*4 as a 



gradually increases. It is easy to give similar expressions for other assumed laws of 



relation of stress to distortion ; but, as will be seen later, this part of the inquiry has not 



yet led to any result of value. 



In testing the results obtained with the earlier apparatus I assumed the force (for the 



more violent impacts) to be as the square of the distortion simply. This gives, in the 



notation of the Tables below, 



D«T- ! oc Hi 



Of course any investigations, based on such simple assumptions as those made above, 

 can be only very rough approximations, since they ignore altogether the true nature of 

 the distortion of either of the impinging bodies, as well as the internal wave disturbance 

 which is constantly passing to and fro in the interior of each ; part of it, no doubt, 

 becoming heat, but another part ultimately contributing to the resilience. In such cir- 

 cumstances the impact may 'perhaps sometimes consist of a number of successive collisions ; 

 certainly the pressure between the two bodies will have a fluctuating value. 



Measurements of the Tracings, and their Reduction. 



From the tracing for each separate experiment the following quantities were carefully 

 determined. Their values are given in the subsequent Tables, under the corresponding- 

 letters below. 



1. Number of vibrations of the fork corresponding to one-sixth of a complete 

 revolution of the disc ........... N. 



Three diameters of the disc were drawn, making angles of 60° with one another, and 

 the number of undulations of the fork-tracing intercepted between each pair of radii was 

 counted. This process was preferred to the simpler one, of counting the undulations in 

 the entire circumference, for two reasons : — it tests the uniformity of the rotation, or a 

 possible shrinking of the photographic paper ; and it makes one common process of 

 measurement applicable to complete traces, and to others which from some imperfection 

 of adjustment presented only parts which were sufficiently distinct. When only one 

 measurement is given under this head, it means either that only one was possible or that 

 ;ill six gave the same result. When two are given, they are chosen as the least and 

 greatest of the six. They usually differ by a small quantity only, and may indicate 



