444 hersey: vibrations op elastic systems 



tion requires that their numerical values be made equal. There 

 were some persons who took the trouble to construct a certain 

 tuning fork of invar, in hopes of diminishing its temperature 

 coefficient ; as would be seen from the above equation, this only 

 made matters worse. 



4. Shrill tuning fork. For any case of pure twisting or shear- 

 ing, such as the very short stubby forks used for producing in- 

 audible notes, a cannot enter (4); c = — 1, and h = | (/3 + 7). 



5. Telephone diaphragm. For the case of a thin flat circular 

 disc clamped at the edge, and so approximating a telephone 

 diaphragm, the expression for c which would be obtained from 

 model experiments in accordance with (4), if it were not already 



available as a result of integration, 3 is c = . With o-=0.3, 



1—0" 



c = about f, so that (4) becomes n = If a — f 13 + i y. Com- 

 plete compensation would require that 13 be about twice a. 



(Steel samples have been found, with - considerably greater 



a 



and also less than 2.) But note from (4) that the a and (3 

 terms will enter with opposite signs whenever C is positive. 

 Since C is positive for the disc shape, it appears that the tempera- 

 ture coefficient of a disc will always be small compared with that 

 of a tuning fork of the same material, no matter what that 

 material is. 



6. Spring with distributed mass. If the mass is uniformly dis- 

 tributed in the direction of the displacement and remains so 

 during the deformation, it may be shown by integration that 



V-f 1 



lml2Tr 



This expression is consistent with 



(6); f(a) being a constant depending on the ratio of the mass of 

 the spring, ra , to the attached mass, m. 



3 Rayleigh in his Theory of Sound, vol. I. § 221 a, gives for the disc a formula 



VLEY AM, -1 



— 0.8331 —J (1 — o- 2 ) 2 



Here L is the diameter and h the thickness. Thus * (<r) = const. (1 — cr 2 ) - *. 

 Differentiating as directed by (4) gives the expression for c. (In footnote 6 of 



32 

 previous paper replace 32 w by ~7r.) 



