Imperfectly Homogeneous Elastic Solid. 245 



of the expression originally given for Ait, we use the second 



approximation, 



2 

 U 



\ (Ai- cZy <iz/ 2 \ dx dy dz J 



But, to pass on, the state of strain of the element depends now 

 upon the following groups of magnitudes: — 



(I.) A,B, C,D,E,F; 

 (II.) wi, sr 2; ot 3 ; 

 (III.) a, b, c, d, e,f; 

 (IV.) v\,v' 2 ,^ 3 ; 

 {Y , clA dA dF 



We may now follow Green, and assume that the energy in 

 the element-volume can depend only on these five groups. It 

 must, however, be independent of group II. ; for if any term 

 contained w ly the energy in the substance might be increased 

 or diminished by giving it a rotation as a whole round the 

 axis of x — a result contrary to experience. It must also be 

 remembered that the constituents of the last three groups bring 

 with them coefficients involving as a factor the intermolecular 

 distance nt ; and this will cause the coefficients of these terms 

 to be very small compared with those which arise from the 

 combination of group I. with itself. 



Designating, therefore, as of class (I., III.) those terms 

 which arise from the combination of groups I. and III., we need 

 only consider the classes (I.,L), (I., III.), (I., IV.), (L,V.) and 

 neglect the classes (III., IV.), (III., V.), (IV., V.), the terms 

 of which would involve, in the case of transmission of a plane 



wave of length X, the factor ( — ) . We shall thus obtain a 



first approximation to the theory of a hetero-homogeneous 

 solid. 



Specification of the Internal Sfress. 

 4. In the case of a perfectly homogeneous elastic solid, the 

 stresses upon the faces of a rectangular element which inter- 

 sect at a solid angle are resolvable, in the first place, into nine 

 forces, S w , 8*y, . . . S**, wherein S MU is the force per unit area 

 on the face perpendicular to the axis of u acting in the direc- 

 tion of the axis of v, and reckoned as tractions exerted by the 

 surrounding substances. And in order that the angular acce- 

 lerations of the element may be all finite, it is necessary that 



