346 Prof. L. Natanson on 



Referring to equation (1), § 4, we see that 



dp ^.'bpde ~dp d^ ^oSJr ,'dpdoc.,'dp_d^ + 'bpdy ,~ 

 dt ~de dt ~d<$> dt B^ dt "d*dt "dfidt ~dydt' V 



By well-known considerations which are always applicable 

 to the case of an isotropic substance, the form of this equation 

 is easily simplified. We find : 



'dp == 'dp = ^P "1 

 Be B<£ Bt 



r • • • • (?) 

 |P=0, |£=0,|H=0. I 



B« op oy J 



Let — h be the value of the first three expressions; using 

 equations (3), equation (2) becomes 



t—* (4 > 



The simplifying hypothesis which we have made use of 

 may also be derived from another hypothesis, resembling that 

 adopted by Sir G. G. Stokes in his theory of viscosity. Let 

 us suppose, in fact, that the pressure p does not vary when 

 the sum e+f+y remains =0 : 



co=e+f+g = (5) 



Thus 



f =° (*> 



for all values of the variables e s f, g, a, b, c which satisfy 

 equation (5). If, for example, 



e=-(f+ff), (7) 



then 



l=(|-|f)^(^-i?> + S a+ l ;+ l c = ' • (8) 



and in this equation the values of f,g, a, b, c are quite arbitrary; 

 thus the proposition under discussion (which is expressed by 

 equations (3) and (4) ) is proved. 



The hypothesis which we have made use of is equrvalent to 

 saying that the pressure p cannot change unless the density 

 of the fluid changes. In order to develop our analysis- 

 further, we shall assume the correctness of this proposition to 

 be a consequence of the following two hypotheses : (1) the 



