for Differential Equations of Mathematical Physics. 595 



The usual process is to eliminate the pressure terms from 

 these equations, giving an equation in yfr alone in the form 



3# B// d* 3,y3# 3#oy 



This equation, derived by differentiating (4) and (5), is one 

 order higher than each of the original, and therefore an 

 equivalent system to (4) and (5) must associate with (6), an 

 equation of one order lower, otherwise extraneous problems 

 may be introduced not contemplated in the discussion. These 

 extraneous cases may be excluded by the following process. 

 Let A and B be two points situated in the fluid, the extremities 

 of any given curve ; then from equations (4) and (5), 



\ (±d,v + Ydy)-- 3 (p A -p B ) = I ^(udx + vdy) 



- v\ [\/hulx + \7 2 vdy). ...... (7) 



Equations (6) and (7) are the equivalent of (4) and (5). 

 Equation (7) is of course an expression giving the pressure 

 difference between two points A and B, and as such need not 

 be considered in relation to (6) except in so far as it must be 

 used for the interpretation of any externally applied con- 

 ditions. It will then furnish information with which the 

 solution of (6) must be consistent. For example, if points A 

 and B are maintained through some external agency at a 

 constant difference in pressure P, then equation (7) will 

 furnish a condition which must be satisfied by the solution 

 of (6) along every curve, lying wholly in the fluid, that can 

 be drawn between A and B. The necessary and sufficient 

 condition that the external forces can only be of a conservative 



nature is that 1 [K.dx + Ydy) should be independent of the 



path. It follows in that case that the path from A to B 

 may be arbitrarily chosen, and one may be selected in any 

 convenient manner, the integration along all other paths 

 reconcilable to this being the same. In general it will 

 be found most convenient to select as a portion of the path 

 the contour of the boundary along which the velocities are 

 specified. 



Consider a contour completely enclosing one of the 



