O.V TilE PLOW OF WATKtt THKOUGH ORIFICES. 243 



Improved Investigations on the Flow of Water through Orifices, with 

 Objections to the modes of treatment commonly adopted. By Prof. 

 James Thomson, LL.D., D.Sc. 



[A communication ordered by the General Committee to be printed in extenso among 



the Keports.] 



The methods usually put forw^ard for treating of the flow of water out of 

 vessels b}- orifices in thin plates, slightly varied though they may be in different 

 cases, are ordinarily founded on assumptions largely alike in these different 

 cases, and largel}' erroneous. The theoretical views so arrived at, and very 

 generally promulgated, are in reality only utterly false theories based on sup- 

 positions of the flow of the water taking place in ways which are kinemati- 

 caU}' and dynamically impossible, and are at variance with observed facts of 

 the flow, and even at variance with the facts as put forward by the advancers 

 themselves of those theories. The admittedly erroneous results brought out 

 through those fallacious " theories," and commonly miscalled " theoretical 

 results," are afterwards considerably amended by the introduction into the 

 formulas so obtained of constant or variable coefficients, or otherwise, so as 

 to be brought into some tolerable agreement with experimental results. 

 These means of practical amendment, however, being themselves not estab- 

 lished on any scientific principles, can at best only conduce to the attainment 

 of useful empirical formulas, but cannot, by their application to the orio-i- 

 naUy false theoretical views, come to develop any true scientific theory. A 

 theory may, no doubt, be regarded as a good scientific theory, and as bein» 

 good for practical purposes, which leaves out of account some minor features 

 or conditions of the actual facts. In so far as it leaves any influential 

 elements out of account, it is imperfect ; but if the conditions which, for 

 simplicity, or from want of complete knowledge of the subject, or for any 

 other reason, are left out be of very slight influence on the practical results 

 in question, the theory may be regarded as a very good one, though not quite 

 perfect. In the case, however, of the hydraulic theories now referred to, the 

 false principles involved in the reasonings relate to the main and important 

 conditions of the flow, and not to anj' mere minor considerations, the imper- 

 fections or errors of which might be of but slight importance in the develop- 

 ment of the main principles involved, and but little influential on the results 

 sought to be attained. 



I will now proceed to give some examples or sketches of the usual 

 methods of treating the subject. 



I will first take the case of water flowing from a state of rest through an 

 oiifice in a vertical plane face. This case is ordinarily treated bj' supposing 

 the orifice to be divided into an infinite number of infinitely narrow horizontal 

 bands of area, and supposing the velocity of the water in each l)and to be 

 that due, through the action of gravity, to a fall from the still-water smi'ace- 

 level down to that band ; then multiplying that velocity by the area of the 

 band, and treating the product as being the volume flowing per unit of time 

 across that horizontal band or element of the area ; and integrating to find 

 the sum of all these volumes of water for all the bands, and treating this 

 sum as being the " theoretical" volume per unit of time flowing across the 

 whole area of the orifice. This result is commonly called the " theoretical 

 discharge " per unit of time ; but, as it is known not to be the actual dis- 

 charge, it is then multiplied by a numerical coefficient called by some " the 



r2 



