128 Mr. Challis 07i the Geiieral Equations 



at a constant level AMD, and let the fluid issue from a cir- 

 cular orifice BC, through the centre of which the axis passes. 



ku 



There must be a very slender column of fluid, having its 

 axis coincident with M N, the particles in which move entirely 

 in a vertical direction, because there can be no reason why 

 they should move in one horizontal direction rather than an- 

 other. To this column, even if it be supposed to be conti- 

 nued out of the vessel, the preceding equation will apply with 

 exactness, because q being =/{t), we need not have regard 

 to the law of continuity in the values of q. This being pre- 

 mised, let k = the area of the orifice B C, O P = x, u = the 

 velocity at N, j = that at P, and z an arbitrary function of 



X, such that q = 



Then 



And — = q, also dY = g dx, V = gx 



]) kx du k- u'' /, xrfiX , T7>; /,\ 



f ■= z d I z- \ - zdx/ ^ 



Let ::' = the area of the upper surface of the fluid. Then 



at M the velocity = -;^', because the vertical velocity at tlie 



surface must be less than that at the orifice in the ratio of 



k to 



