ON THE FLOW OF WATER THROUGH ORIFICES. 217 



must be zero ; and that component, not the velocity itself, is what the 

 ordinate of the true curve must represent. On the hj'pothesis of perfect 

 fluidity in the water (which, throughout the present discussions and investi- 

 gations, is assumed as being a close enough representation of the truth to 

 form a basis for very good theoretical views), the velocities at top and bottom 

 of the orifice will be those due by gravity to falls from the still- water surface- 

 level down to the top and bottom of the orifice respectively, because at these 

 places the water issues really into contact with the atmosphere, and 

 consequently attains atmospheric pressure. At all intervening points in the 

 plane of the orifice it maj' readily be seen, or may with great confidence be 

 admitted, that the pressure will be in excess of the atmospheric pres- 

 sure ; because, neglecting for simplicity the slight and, for the present 

 purpose, unimportant modification of the courses of the stream-lines caused by 

 the force of gravity acting directly on the particles composing the stream- 

 lines, as compared with the courses which the stream-lines would take if the 

 action of gravity were removed, and the water were pressed through the 

 orifice merely by pressure applied, as by a piston or otherwise, to the fluid 

 in the vessel, we may say, truly enough for the present purpose, that an 

 excess of pressure at the convex side of any stream -line is required in order 

 that the water in the stream-line can be made to take its cui-ved path. The 

 mode of reasoning on this point suggested here may be obvious enough, 

 although, for the sake of brevity, it is here not completely expressed. It fol- 

 lows that at all these intervening points in the plane of the orifice the absolute 

 velocity of the water will be less than that due to a fall from the still-water 

 surface down to the level of the point in the orifice ; and besides, at aU depths 

 in the plane of the orifice except a single medial one, the direction of the 

 flow will be oblique, not normal, to the plane of the orifice. Hence, 

 further, through these two circumstances, jointly or separately as the case may 

 be, it follows obviously that the ordinates of the true curve will everjTvhere 

 be less than those of the parabola. 



Fig. 4 illustrates in like manner the false theoretical and the true actual 

 conditions of the flow over a level upper edge of a vertical plane face, which 

 may be exemplified by the case of a rectangular notch without end contrac- 

 tions, or of a portion of the flow not extending to either end in a very wide 

 rectangular notch. In this case it is to be observed that the ordinates at 

 and near the top of the issuing Avater in the vertical plane of the orifice 

 must be only slightly less than those of the parabola — because, at the very 

 top or outside of the stream, atmospheric pressure is maintained throughout 

 the length of any stream-line, and so the velocity will be very exactly that 

 due by gravity to the vertical depth of the flowing particle below the still- 

 water surface-level in the vessel ; and because, also, the direction of the 



contracting vein of water issuing from a circular orifice in a thin plate, a solid of revolu- 

 tion specilied clearly in such a way that the water surface in leaving the plane of the plate 

 makes an angle of about 67° with that plane, and states to the effect that that water sur- 

 face is just a continuation of the paths of the stream-lines within the vessel which he 

 represents at the margin of the orifice as crossing the plane of the orifice with conrerging 

 paths making the angle already mentioned of about ti7° with that jslane. They ought in 

 reality to leave the lip tangentially to the plane, and then to make a very rapid turn in a 

 short space (or to have a very small radius of curvature) on just leaving the lip of the 

 oiiflce. The prevalence of erroneous representations and notions on this subject was 

 adverted to, and an amendment was adduced, by myself in a Report to the British Asso- 

 ciation in 1861 on the Gauging of Water by V-Notches (Brit. Assoc. Rep. Manchester 

 Meeting, 1861, part 1, p. 156). 



