ON THE FLOW Ol' WATER THROUGH ORIFICES. 245 



the same dii'ectiou at diiferent parts ; and the assumption of the velocity 

 being the same throughout, together with the assumption tacitly implied of 

 the direction of the motion being the same throughout, vitiates the reasoning 

 very importantly. It is thus to be noticed at the outset that the division of 

 the orifice into bands, infinitely narrow iu height, but extending horizontally 

 across the entire orifice, cannot lead to a satisfactory process of reasoning, and 

 that the elements of the area to be separately considered ought to be infinitely 

 small both in length and in breadth. 



iiecondli/. For anj' element of the area of the orifice infinitely small in 

 length and breadth it is not the velocity of the water at it that ought to be 

 multiplied by the area of the element to find the volume flowing per imit of 

 time across that element, but it is only that velocity's component which is 

 normal to the plane of the element that ought to be so multiplied. 



Thirdly. Whether, for any element of the area of the orifice, we wish to treat 

 of the absolute velocity of the water there, or to treat of the component of that 

 velocity normal to the plane of the orifice, it is a great mistake to suppose 

 that the velocity at the element is that due by gravity to a fall from the still- 

 water surface-level of the pent-up statical water down to the element. The 

 water throughout the area of any closed orifice in a plane surface, with the 

 exception of that flowing in the elements situated immediately along the 

 boundary of the orifice, has more than atmospheric pressure ; and hence it can 

 be proved * that it must have less velocity than that due to the fall from the 

 still-water surface-level down to the element. 



The foregoing may be illustrated by consideration of the very simple case 

 of water flowing from a vessel through a rectangular orifice in a vertical 

 plane face, two sides of the rectangle being level, and the other two vertical, 

 and end contractions being prevented by the insertion of two parallel guide 

 walls or plane faces, one at each end of the orifice, and both extending some 

 distance into the vessel perpendicularly to the plane of the orifice, so that 

 the jet of issuing water may be regarded as if it were a portion of the flow 

 through an orifice infinitely long in its horizontal dimensions. 



Thus if the jet shown in section in fig. 3a be of the kind here referred to, 

 while W L is the still- water surface-level, the so-called " theoretical velocities" 

 at the various depths in the orifice, which are dealt with as if they were in 

 directions normal to the plane of the orifice, can be, and very commonly are, 

 represented by the ordinates of a parabola as is shown in fig. 36, where B D 

 represents in magnitude and direction the "theoretical velocity" at the top of 

 the orifice, C E the " theoretical velocity " at the bottom of the orifice, and 

 F G that at the level of any point F in the orifice — these ordinates being each 

 made = ^2gh, where h is the depth from the still- water surface down to the 

 level of the point in the orifice to which the ordinate belongs. Then, under 

 the same mode of thought, or same set of assumptions, the area of that 

 parabola between the upper and lower ordinates (B D and C E) will represent 

 what is commonly taken as the " theoretical discharge " per unit of time 

 through a unit of horizontal length of the orifice. But this gives an exces- 

 sively untrue representation of the actual conditions of the flow. Instead of 

 the parabola, some other curve, very different, such as the inner curve 

 sketched in the same diagram, fig. 3h, but whose exact form is unknown, 

 would, bj- its ordinates, represent the velocity-components normal to the 

 plane of the orifice for the various levels in the orifice, and its area would 

 represent the real discharge in units of volume per unit of time through 



* Theorem I., further on, will aflbril proof of this. 



