Prof. Challis on the Principles of Hydrodynamics. 31 



at any time t, p the pressm-e, and p the density at that position, 

 and let X, Y, Z be the impressed accelerative forces. Since by 

 Prop. II. the law of equal pressui-e in all directions from any 

 position is true of fluid in motion, it may be shown, by proceed- 

 ing as in Prop. III., that the accelerative forces of the element 

 due to the pressure and tending towards the coordinate planes, are 

 dp dp dp 

 pdx' pdif pdz' 

 By impressing continually on the whole fluid motions equal and 

 opposite to the actual motions of the element, the above forces 

 remain the same, and the element is reduced to rest. But, the 

 element being at rest, these forces must be in equilibrium with 

 the impressed accelerative forces, which are now 

 d^x dh, dH 



^~ dt^' dt^' dt^' 



Hence, as in Proposition III., 



f = (x-$)^.(y-£^)...(z-J).. . (..) 



where dx, dy, dz are independent variations of the coordinates. 

 This is the requii-ed general equation. Being proved to be true 

 of any. element at any time, it may be extended to all parts of the 

 fluid at all times, provided certain equations of continuity, to be 

 presently investigated, are satisfied. 



Corollary. Let u, v, w be the components, in the directions of 

 the axes of coordinates, of the velocity (V) which exists at any 

 point xyz at any time t. Then if x + dx, y + dy, s + ds be the 

 coordinates at the time t + dt oi the element which was at xyz at 

 the time t, we shall have 



dx dy dz 



dt' '- "'- 



And since, generally, 



(rfwX _ du 

 dt)~'di 



it follows that 



d'^x du du, du du 



W=Tt^Tx''''lbj''-^Tz'"- 

 So 



«pw dv dv dv dv 



J = di^Tx''-^-¥/-^dz''' 

 and 



d!^z _dw dw dw div 



di^ ~ dt dx dy dz ' 



which values are to be substituted in the equation (1.). 



^=^' ^=¥' ^=^' 



(du\ _ d'^x 

 It) "IF' 



iu du dz 

 , . _i J . 



dx' dt "^ dy dt dz dt ' 



du dx du dy du dz 



