the Laws of Viscosity. 355 



shown to be equivalent to 



*>**-(* (2) 



provided we suppose /ul ^ 0, which is legitimate. 



Let us now return to the study of the two constants of 

 viscosity introduced into our theory by means of equations 

 (14) and (15) of the preceding paragraph. These equations 

 teach us that the relation X= — |yu,, proposed by Stokes and 

 accepted by the majority of scientists, is an immediate con- 

 sequence of the equation h = k, of which we have given a 

 detailed discussion above. If, on the other hand, we suppose 

 that h and k may be unequal, then 



and the relation connecting X and /jl will depend not only on 

 the ratio k/n, but also on that of the new constant h to the 

 rigidity n. 



As to the inequality (2), the only consequence which may 

 be deduced from it is that 



***, (4) 



and this new inequality is certainly verified in the case of 

 actual fluids. 



In conclusion, we may say that the equation h=k, and 

 Stokes's relation, \=— |/u, agree perfectly with the whole 

 of our hypotheses ; but there is nothing to lead us to regard 

 them as necessary corollaries of our theory. 



§ 10. Let X, Y, Z be the components, per unit of mass, 

 of the external force which acts on an element of volume at 

 the point (x % y, ~). Then we have three equations, the first 

 of which is 



p g? + u !" + , |? + w p) = pX - &s + ^ + &s) (U) 

 r \^t d# d# o~. ; r \d<- dy 3« / 



From this, taking into account equations (16) and (17) 

 of §8, 



= p X~^~e-'rr(^+^+^) y. (2fl) 



r do? \ d# oy o~ / v ' 



In this equation the symbol V 2 represents the well-known 



