174] ISOTBOFIS^]^Bp i1 ^gJg^TIAL. 459 



which is the form used by Kirchhoff. We have then for the rela- 

 tions between the constants, 



136) H-C = 2K and 3C=2K 



or 



(1 \ 9 



& _i_ _\ n JC 

 3 / 3 



Accordingly for liquids in order to have (7 = but H finite, we 

 must put K= and = oo, so that 2K = J7. 

 Now since 



J.O < ) A< ~j~ A2 ~j" Ag === -tj ^ -*-2 



f _L __L ^2 O ' /" j_ i_ 2 ^2 2\ 



we have for solids, liquids and gases, 



138) = K0* + K{(s* + sl + sl) 



We shall make use of the more common notation 



139) 2K& = 1, K=ii. 



(Thomson and Tait make use of the constants k for H and n for /i.) 

 We have accordingly 



140) $ = Ji(s x + s v + s,y 4- ii{sl + s 2 -f si + 2(^ 2 + ^ + <7 2 )} 



- P (S,,. + S y + S,). 



The constant ^ like (7 refers to changes of form and vanishes for 

 perfect fluids. In the present notation by equations 136) we have 



141) H = ^ + l, 



so that both >L and p are involved in changes of volume. We thus 

 see that isotropic bodies possess two elastic constants. By means of 

 certain assumptions as to the nature of elastic stresses, making them 

 depend upon actions between molecules, Cauchy and the earlier 

 writers on elasticity reduced the energy function to a form depending 

 on a single elastic constant, the same theory reducing the number 

 of constants for an eolotropic body from twenty- one to fifteen. For 

 this theory the reader may consult Neumann, Theorie der Elastizitat, 

 Todhunter and Pearson, History of the Theory of Elasticity. Experiments 

 have not however confirmed this theory, and it is no longer generally 

 held to be sound. Thomson and Tait inveigh against it with particular 

 emphasis. We shall accordingly assume that an isotropic body has 

 two independent constants of elasticity A and ^. 



