A FLUID UNDER CONSERVATIVE FORCES. 171 



+ A{(Scrv)</) + A'(So■v)^^J 

 = V .B^ + kv .D,-»/r + ivo-^ 

 and, finally, the equation of motion is 



Now this equation is equivalent to three scalar equa- 

 tions; and if we make in it the substitutions D^yir^o, 

 Dtyjr = o, we reduce it to a single equation and solvable ; 

 for then 



vD,0 = -v(v + | + -i), 

 or 



m 2 



where V~'o is independent of the position and may be 

 included in the (f> on solution. 



Putting for Dt<p, <f) — {SaV)4>, and substituting for a 

 we get 



or 



2<^-(V</>)-^ + /:^(V^)^=-2V-^ (IV.) 



In support of the substitutions Dtk=o and D<'v/r=o I 

 should state 



( 1 ) That such surfaces can be found from the differential 

 equation. 



(2) That only three scalar equations have been used in 

 determining a so as to satisfy the equation of motion. 



(3) That as the intersections of such surfaces, if they 

 exist, arc to move with the fluid, it is not imnatural to 



