350 G. F. Becker — Finite Elastic Stress-Strain Function. 



musicians nor physicists have detected any such variation of 

 pitch which, if sensible, would render music impossible. The 

 fact that the most delicate and accurate microchronometrical 

 instruments yet devised divide time by vibrations of forks, is 

 an additional evidence that these are isochronous. Lord Kelvin 

 has even suggested the vibrations of a spring in a vacuum as 

 a standard of time almost certainly superior to the rotation of 

 the earth, which is supposed to lose a few seconds in the 

 course of a century.* 



It is therefore a reasonable hypothesis in the light of experi- 

 ment that the load strain function is such as to permit of iso- 

 chronous vibrations ; but to justify this conclusion from an 

 experimental point of view, it must also be shown that Hooke's 

 law is incompatible with sensibly isochronous vibration. I 

 shall therefore attempt to ascertain what load -strain function 

 fulfills the condition of perfect isochronism (barring changes 

 of temperature) and then to make a quantitative comparison 

 between the results of the law deduced and those derived from 

 Hooke's law. 



Application of moment of momenta. — If the cube circum- 

 scribed about the sphere of unit radius is stretched by oppos- 

 ing initial stresses and then set free, it will vibrate ; and the 

 plane through the center of inertia perpendicular to the direc- 

 tion of the stress will remain fixed. Each half of the mass 

 will execute longitudinal vibrations like those of a rod of unit 

 length fixed at one end, and it is known that the cross section 

 of such a rod does not affect the period of vibration, because 

 each fiber parallel to the direction of the external force will 

 act like an independent rod. Hence attention may be con- 

 fined to the unit cube whose edges coincide with the positive 

 axes of coordinates, the origin of which is at the center of 

 inertia of the entire mass. 



The principle of the moment of momenta is applicable to 

 one portion of the strain which this unit cube undergoes dur- 

 ing vibration. The moment of a force in the xy plane rela- 

 tively to the axis of 02, being its intensity into its distance 

 from this axis, is the moment of the tangential component of 

 the force and is independent of the radial force component. 

 Now dilation is due to radial forces and neither pure dilation 

 nor any strain involving dilation can be determined by discus- 

 sion of the moments of external forces. Hence the principle 

 of the moment of momenta applies only to the distortion of 

 the unit cube. This law as applied to the xy plane conse- 

 quently governs only the single shear in that plane. 



The principle of the moment of momenta for the xy plane 

 may be represented by the formula 



* Nat. Phil., sections 406 and 830. 



