124 Mr. TV. M. Buchanan's Theory of the Ileaction Water- Wheel. 



ascribed to the inclined directions which the molecules of the fluid assume 

 previous to their exit, and which they tend to retain after passing the 

 thin parietes of the simple orifice. For greater clearness let us assume 

 that the aperture is horizontal, circular, and of small area in comparison 

 with the area of the containing vessel ; under these conditions a large 

 portion of the fluid will be put in motion, and will slowly approach the 

 orifice during the efflux, in the form of an inverted cone, of which the 

 orifice is the apex. The particles, as they come opposite to the orifice, 

 are therefore impressed with motions converging to an axis ; but these 

 motions, in consequence of the mutual cohesion of the particles, must tend 

 to a common velocity in that axis : and the length of the external conoid 

 will express the time in which the oblique motions are converted into 

 motions parallel to the axis of the jet. It is therefore only at the point 

 of least section that the molecules of fluid have attained the effective 

 velocity due to the head under which they issue ; and it is therefore only 

 in reference to that point that the hydraulic pressure of the jet is equal 

 to a column of the fluid of double the actual liead. By adopting an 

 ajutage to the orifice of the shape indicated, the oblique motions of 

 the particles are corrected in passing through it, and reduced to parallelism 

 with the axis at the moment of efflux into the atmosphere. There still, 

 however, remains to depreciate the discharge assigned by the formula 

 <7 = aV2<7H, the imperfections of workmanship in the construction, 

 and the adhesion of the fluid to the perimeter of the ajutage, with possibly 

 a slight atmospheric influence not yet defined. But assuming the ajutage 

 to be made with all possible care — both as to form and finish — if we call 

 the area of the orifice 1000, that of the contracted vein will be 975 : 

 and these numbers taken inversely will express the velocity of the jet at 

 the two points measured by the discharge. The value of q for an orifice 

 of this form will therefore be 



2=-975aV2#H 

 showing a loss of head-pressure, as measured by the discharge, of 



(1 — -9T5 2 ) — = 049375 H 



when U = V 2g H the theoretical velocity due to the head H. And 

 generally, if V be the actual velocity of efflux, and h the practical coeffi- 

 cient of discharge for any orifice, so that U = -j- ' the head-pressure not 



realized in the measure of o, will be (-— — 1 ) — -— = °— — And the 



VF / 2g 2g 



pressure not realized in the measure of the reaction, will be expressed by 



I V 2 V 2 



in which <P denotes the mean angle formed by the filaments of water of 

 the jet with the axis. 



