The Aether as an Elastic Solid. 163 



then yields the differential equations of motion, namely : 



8 / de x de y fe,\ a / ae x a^ , a^\ 



+ a + h ^ + g + a + A + ], 

 acVSaj ty y dzj dx\ fa ty y fa) 9 



and two similar equations. 



These differ from Cauchy's fundamental equations in having 

 greater generality: for Cauchy's medium was supposed to be 

 built up of point-centres of force attracting each other according 

 to some function of the distance ; and, as we have seen, there 

 are limitations in this method of construction, which render it 

 incompetent to represent the most general type of elastic solid. 

 Cauchy's equations for crystalline media are, in fact, exactly 

 analogous to the equations originally found by Navier for 

 isotropic media, which contain only one elastic constant instead 

 of two. 



The number of constants in the above equations still exceeds 

 the three which are required to specify the properties of a 

 biaxal crystal : and Green now proceeds to consider how the 

 number may be reduced. The condition which he imposes for 

 this purpose is that for two of the three waves whose front is 

 parallel to a given plane, the vibration of the aethereal molecules 

 shall be accurately in the plane of the wave : in other words, 

 that two of the three waves shall be purely distortional, the 

 remaining one being consequently a normal vibration. This 

 condition gives five relations,* which may be written : 



a. b = c = JJK; 



/'-j-2/ / = M -2<7; tf- M -2fc; 

 where /z denotes a new constant, f 



* As Green showed, the hypothesis of transversality really involves the existence 

 of planes of symmetry, so that it alone is capable of giving 14 relations between the 

 21 constants : and 3 of the remaining 7 constants may be removed by change of 

 axes, leaving only four. 



t It was afterwards shown by Barre de Saint- Venant (b. 1797, d. 1886), 

 Journal de Math., vii (1863), p. 399, that if the initial stresses be supposed to 

 vanish, the conditions which must be satisfied among the remaining nine constants 



M 2 



