1«6 Mr CHALLIS's RESEARCHES IN THE THEORY 



and differentiating under the sign /, ^ = f -r-^^ +J''(^)- Hence 

 substituting for -^ in the known expression for the pressure {p), 



p = f(Xdx + Ydy + Zdz) - f^ds - ^ -fit). 



If f^ be always the same in quantity and direction at the same point, 



dr V^ 



-^ = : so that, p = j{Xdx + Ydy + Zd%) - -— -f{t). 



This equation may thus be considered to be strictly deduced from the 

 general equations of fluid motion. 



Considerations analogous to those applied (Arts. 6 and 7) to motion 

 in a plane, might here be introduced to shew that the motion at any 

 point, when due to several causes, is the resultant of the motions which 

 would be produced by the causes acting separately ; and also to determine 

 the same law of reflection at a plane surface. 



12. The following simple instance of fluid motion may serve to 

 illustrate some points of the preceding theory. BCD (Fig. 2.) is a 

 conical vessel with its axis vertical. A mass of fluid DBhd is made 

 to descend so that its lower surface hd is bounded by a horizontal 

 plane to which any arbitrary velocity is given. The upper surface is 

 also supposed to be plane and to be kept horizontal by the force of 

 gravity. It is required to find the consequent velocity and pressure 

 at every point of the fluid. 



It is evident that the motion will be in vertical planes passing through 

 the axis, and will be, the same in all such planes. Take therefore two 

 planes making an indefinitely small angle with each other, and let 

 AB, AE, be their intersections with the upper surface, ab, ae, with 

 the lower. Let PQSB be an element of the upper surface, P and B 

 being equidistant from A, as also Q and S. If now at any instant 

 lines commencing at the four points P, Q, B, S, be continually drawn 



