hersey: vibrations of elastic systems 439 



The first two results come at once from dimensional reasoning on 

 the assumption that the absolute size L, the density p, together 

 with any two of the three elastic constants E, n, and <x, form a com- 

 plete set of independent variables. Those two results, ex cept for 

 the explicit mention of Poisson's ratio, will be familiar to any one 

 who has consulted Lord Rayleigh's writings on Sound. The tran- 

 sition to the last two results is obvious, and these will be more use- 

 ful for some of the present purposes, because they involve a smaller 

 number of quantities that are affected by temperature. This hap- 

 pens because of the physical fact that the mass of a body does not 

 sensibly alter with temperature, whereas both the length and 

 density do. The functions <J> , $, etc., are unknown, but if neces- 

 sary could be determined by model experiments; thus, for ex- 



i — ■ 



ample, <t> would be found by plotting observed values of n L \-^ 



* E 



against known values of a. 



General formula for change in frequency of a single body with 

 temperature. Differentiating the second part of (1) logarith- 

 mically, treating m as a constant, and simplifying mathematically 

 without introducing any further physical information, gives 



n = -— + *(<* + t) 

 $ de 





(2) 

 - -J- + H/3 + 7) 



^ de 



In case the first term vanishes, a simple result is already avail- 

 able. This will happen for pure bending or stretching, for pure 

 twisting or shearing, and for materials, if any such exist, in 

 which Poisson's ratio does not change with temperature. For, 

 with pure bending etc., <t> is constant, so that n = \ (« + 7); 

 while, with pure twisting, etc., ^ is constant, so that n = \ (0 + 7) . 

 In general, however, some physical law must now be made use 



of to evaluate — . 

 dd 



The foregoing results, once it is stipulated which one of the 



multiple values of the several constants is to be understood by a 



given symbol, are perfectly applicable to heterogeneous and 



