G. F. Becker — Finite Flastic Stress-Strain Function. 355 



u= V Q so that V - = a/W+4. 

 4 * pl?i(l+s) u f s 



If s = 0*01, this expression gives v/u — 400/401. 



It would appear then that on the hypothesis of Hooke, a 

 note due to longitudinal vibrations of about the pitch G s 

 would give a lower note when sounding fortissimo than when 

 sounding pianissimo, and that the difference would be one 

 vibration per second, or one in four hundred. But accord- 

 ing to Weber's experiments experienced violin players distin- 

 guish musical intervals in melodic progressions no greater than 

 1000/1001, while simultaneous tones can be still more sharply 

 discriminated.* The value of s corresponding to v/u — 1000/ 

 1001 is only 0*001, and consequently strains reaching only 

 about one-third of the elastic limit of piano wire should give 

 sensible variations of tone during the subsidence of vibrations 

 if Hooke's law were correct. 



Longitudinal vibrations are not so frequently employed to 

 produce notes as transverse vibrations. The quantity M/p 

 enters also into the expression for the frequency of transverse 

 vibrations though in a more complex manner. In the case of 

 rods not stretched by external tension, the ratio v/u would 

 take the same form as in the last paragraph. One theory of 

 the tuning-fork represents it as a bar vibrating with two nodes, 

 and therefore as comparable to a rod resting on two supports. 



A pair of chronometrical tuning-forks could be adjusted to 

 determine much smaller differences in the rate of vibration 

 than 1000/1001 ; for the relative rate of the forks having been 

 determined on a chronographic cylinder for a certain small 

 amplitude, one fork could be more strongly excited than the 

 other and a fresh comparison made. The only influences tend- 

 ing to detract from the delicacy of this method of determining 

 whether change of amplitude alters pitch, would seem to be 

 the difficulty of sustaining a constant amplitude and the differ- 

 ence of temperature in the two forks arising from the dissipa- 

 tive action of viscosity. 



Conclusion. — The hypothesis that an elastic isotropic solid 

 of constant temperature is such as to give absolutely isochron- 

 ous longitudinal vibrations leads to the conclusion In (a 2 A)= 

 Q/M without any apparent alternative. Comparison with the 

 results of Hooke's law shows that, if this law were applicable 

 to finite vibrations, easily sensible changes of pitch would 

 occur during the subsidence of vibrations in strongly excited 

 sonorous bodies. — The logarithmic law is the same deduced in 

 the earlier part of the paper from the ordinary definition of 

 the ideal elastic isotropic solid, based upon experiments on 



*Helmholtz, Tonempfindungen, page 491. 



