SURFACES OF DISCONTINUITY 623 



position and (p' + 8p') — (p" + 8p") must be less than 

 (o- + 5<r) (ci + 8ci + C2 + 5C2), so that 



8{p' - p") < (ci + C2)5<r + a 8(ci + C2). 



As every one of the variables can be represented as a function of 

 v' it follows that, for mechanical stability of the surface, 



djci + C2) dp' dp" da 



" dv' ^ dv' ~ dv' - ^'' + ^^ 'cb'' 



Now it can be shown that 



ds 



where s is the area of the surface, bounded as it is by the edge of 

 the orifice. (See the note on curvature, p. 10 of this volume.) 

 Hence it follows that 



d^s dp' dp" d(T ds 

 '^ d7^ ^ di/ ~ ~d7 ~ d^'"dv'' 



which is just equation [544]. The problem can be completed 

 as on page 251. Thus we see that the same conclusion is 

 reached as before when we took no special heed of mechanical 

 stability and merged that stability, as it were, in the general 

 method of dealing with stability with reference to the neigh- 

 boring equilibrium and quasi-equilibrium states. This provides 

 still further justification for the validity of the general method. 

 The only point of special importance about the problem on 

 page 245 concerns the assumed circularity of the orifice. One 

 then has special values for ds/dv' and d^s/dv'^. These can be 

 derived from the special geometry of the case as outlined in the 

 middle of page 245; by the aid of the equations there one can 

 prove that 



and 



