440 hersey: vibrations of elastic systems 



anisotropic bodies; nor need the vibrations be small enough to 

 be isochronous, provided the restriction to the same generalized 

 shape is taken to include the deformations. But, now, for 

 homogeneous, isotropic bodies 



.-±2-1 (3) 



Differentiating (3) , and substituting the result in (2), gives finally 



n = i[(c+l)a-cp + y] 



in which 



r, /-. , . *' /-, , \ d , fmn 2 \ 

 C = 2(l+a)--= (l + ff )_log (-— J ) (4) 



<J> da \LE/ ( 



= 2(1 + a)- -I.-. (l + ff — logf — J--.1 

 ^ da \ L/jl / 



The simplicity of equation (4) is noteworthy; it shows that, for 

 bodies made of materials for which a, /3, and y are known, the effect 

 of temperature on frequency can be determined empirically with- 

 out altering the temperature of the models; and, therefore, with- 

 out any restrictions on the thermal properties of the materials 



the models are made of. Note 1 that in plotting — - or 



LE L /j. 



against a to determine c as a function of a graphically, relative 

 values are sufficient; for the existence of an unknown con- 

 stant numerical factor will not alter the slope of a logarithmic 

 plot. The same remark applies below, wherever similar expres- 

 sions occur. 



From (4) the condition for temperature compensation is that 



t-l + l(l+l) (5) 



a C \ a/ 



When 7 is small compared to a (for steel 7 is about 1/25 of a), 



- = 1 -| — . The two methods of compensation suggested in 

 a c 



the previous paper — altering the composition of the material, 



or altering the geometrical shape of the body — are equally 



applicable here. 



