8 10.] SURFACE CONDITIONS. 7 



essentially positive, at all events in the case of ordinary fluids, 

 which cannot sustain more than an infinitesimal amount of tension 

 without rupture. Hence if in any of our investigations we be led 

 to negative values of p, the state of motion given by the formulae 

 is an impossible one. At the moment when, according to the form- 

 ula3, p would change from positive through zero to negative, 

 either the fluid parts asunder, or a surface of discontinuity is 

 formed, so that the conditions of the problem are entirely changed. 

 See Art. 94. 



The quantity p is finite and positive, but not necessarily con 

 tinuous. 



10. The equations, which have been obtained so far, relate to 

 the interior of the fluid. Besides these we have, in general, to 

 satisfy certain boundary conditions, the nature of which varies 

 according to the circumstances of the case. 



Let F(x, y, z, t) = ........................ (10) 



be the equation to a surface bounding the fluid. The velocity 

 relative to this surface of a particle lying in it must be wholly 

 tangential (or else zero), for otherwise we should have a finite flow 

 of liquid across the surface, which contradicts the assumption that 

 the latter is a boundary. The instantaneous rate of variation of F 

 for a surface-particle must therefore be zero, i.e. we have 



IT- ............................ &amp;lt;&amp;gt; 



This must hold at every point of the surface represented by (10). 



7 rr 



At & fixed boundary we have r- = 0, so that (11) becomes 



dF , dF , dF 



Uj- + v-j-+w- r - = 0, 

 ax dy dz 



or, if I, m, n be the direction-cosines of the normal to the surface, 



lu + mv + nw = ..................... (12). 



If F= be the equation of a surface of discontinuity, i. e. a 

 surface such that the values of u, v, w change abruptly as we pass 

 from one side to the other, we have 



dF dF dF dF . 



dF dF dF dF A 



and -j- + u. -j- + v. -j- 4- w. -j- = 0, 



dt 2 dx 2 dy * dz 



