344 



Prof. L. Natanson on 



and k for the modulus of compressibility ; these will he ideal 

 values which determine the elastic properties of the medium at 

 a certain instant — that corresponding to t = 0. Letp be the 

 pressure (normal and equal in every direction) which, at the 

 instant considered, would exist at (#, y, z) had no deformation 

 taken place there. The classical theory teaches us that the 

 inequalities of pressure which, at the instant £ = 0, are called 

 into play by the distortion are given by : 



P\-P*= -2?2<£ -(*-l*)A ; 

 p o zz -pO = — 2n^o - (k — hi) A ; 



p**= 



$!>„=— np>; . 



P Q xy=— «7°- 



(la) 



(lb) 

 (lc) 



(2 a) 



(2 b) 

 (2 c) 



Nevertheless, this state of affairs will not persist. Beyond 

 the instant ^ = 0, two phenomena appear. In the first place, 

 we notice modifications which depend on external influences. 

 In the second, as the distortion becomes feebler and the 

 inequalities of pressure tend to disappear, the system under- 

 goes what has been called " relaxation/" as explained above. 



§ 3. The simplest hypothesis which maybe made regarding 

 the action of external influences consists in supposing that 

 this action is, like the initial state, subject to the laws of ideal 

 elasticity. Adopting this supposition, it is easy to see that 

 the effects due to external forces may be expressed as follows: 



(dp xx \ 



V dt/i 



( d Pm 



V dt 



d Pyy \ _ 

 dt 



— 2ne—(k — %n)w, 



— 2nf—{k—2n)co, 



\ dt A 



( d PvA _ 

 V dt 



CM—'. 



2ng—(k—$n)a>, 



• (la) 



• (It) 



• (1«) 



• (2 a) 



(2 5) 

 (2 c) 



