112 DYNAMICAL THEORY OF SOUND 



By solving the equations (1) we can express e l} e 2 , e 3 as linear 

 functions of p lt p z , p a . It is obvious, however, that the formula 

 for e l must involve p z and p s symmetrically; and from this 

 consideration, and from the physical meanings of the constants 

 E and cr, it follows immediately that the result must be 

 equivalent to 



E t *>p l -#(p t +p t ), ; 



............... (9) 



Of the various elastic constants and their combinations, 

 one or other may appear specially important, according to the 

 nature of the question in view, and this may account for the 

 great diversity of notations which has arisen. In any case two 

 independent quantities are necessary and sufficient to define the 

 elastic behaviour of an isotropic substance. From a physical 

 standpoint K and p, might appear to be the most fundamental ; 

 whilst as regards facility of direct measurement preference may 

 be given to E and //,, whence K and cr can be derived by the 

 formulae 



fiE E 



" = 9 P -3E' ^V" 1 ' ............ (1 



which follow easily from (3), (6) and (7). On a particular 

 hypothesis as to the ultimate structure of an elastic solid 

 Poisson was led to the conclusion that the two elastic constants 

 are not independent, but are connected by an invariable relation, 

 which in our notation is expressed by X = /JL. This makes 



=/*, E=ln, cr = J ................ (11) 



On experimental grounds Wertheim (1848) proposed the 

 relation X = 2/A, which makes 



* = f ft E = K, <r = J ................ (12) 



More accurate methods of measurement, introduced by 

 Kirchhoff* and others, support the view, which has been con- 

 sistently held by English physicists f that there is no necessary 



* G. E. Kirchhoff (182487), professor of physics at Heidelberg 185475, 

 at Berlin 1875 87 ; famous for his share in the discovery of spectrum analysis, 

 but the author also of important memoirs on the theory of elasticity and its 

 applications to the vibrations of bars and plates. 



t Notably by Green (17931841), Stokes, and Lord Kelvin. 



