220 VISCOSITY. [CHAP. ix. 



at a temperature of 24 5C, the units being the same as before. 

 The viscosity of liquids diminishes rapidly with the increase of 

 temperature. 



179. Let us calculate the rate at which the stresses acting on 

 a rectangular element dxdydz having its centre at (as, y, z) do 

 work in changing its shape and volume. The two yz-faces give 

 du dv dw 



and combining with this the similar expressions due to the other 

 pairs of faces we obtain 



(p xx a + p yv b + p zz c + 2 

 or, by (11), 



p (a + b + c) dx dy dz 



............... (12*). 



The first line in this formula expresses the rate at which a uniform 

 pressure p works in changing the volume of the element at the 

 rate a + b + c. Since this term changes sign with a, b, c, it follows 

 that if p be a function of the density, (and therefore of the volume 

 of the element,) the work done by p during a compression is re 

 stored during an expansion, and vice versa^&quot;. The expression in 

 the second line of (12), however, does not change sign with a, 6, c, 

 and. represents a real loss of energy. The internal friction there 

 fore involves a dissipation of energy at the rate F per unit volume, 

 where 



F may be called, as it is by Lord Rayleigh J, the dissipation- 

 function. We notice that 



Pv. = i ~ d j i &c -&amp;gt; &c - 



* Stokes, Camb. Trans. Vol. ix. p. 58. 



t The assumption that the intrinsic energy of an element of (perfect) fluid 

 depends only on its volume is the basis of Lagrange s treatment of Hydrodynamics 

 in the Mecanique Analytiquc. 



Proc. Lond. Math. Soc. June 12, 1873. 



