ON THE FLOW OF WATEK THROUGH ORIFICES. ^^59 



filaments, while the external filaments would fail to exert that necessary 

 confining pressure. These external filaments could, with very little change 

 in their own velocities, allow even of a great augmentation of the cross- 

 sectional area of the jet if the internal filaments, by abated velocity, were 

 requiring to become considerably thicker fhuu before, in virtr.e of the intro- 

 duction of the obstruilifin. It is only the mpid change of direction of motion 

 of the particles of water in the outer filam.enta in the neighbourhood of G, 

 close to the obstruction, that enables them, by what may be called their 

 centrifugal force, to maintain a greatly increased internal pressure very close 

 to the obstruction, and so to allow of Ihc water in the internal stream-fila- 

 n;ents abating its velocity, and of tliosc filaments themselves swelling in their 

 transverse dimensions. 



These considerations complete all that is necessary for the demonstration 

 of Theorem II., and it may now be regardtd as proved. 



FoRMTTLA FOB THE FlOW OF "WaTER IN THE V-NOTCH. 



From the foregoing principle we can find intuitively the formula for the 

 quantity of water which will flow through a V-notch in a vertical plane sur- 

 face, as in fig. 11. We can see it at once by considering any stream-filament 



Fig. 11. 



in the flow in one notch, and the homologous stream-filament in the simila 

 flow in another notch similarly formed, but having its vertex at a difl'erent 

 depth below the still-water surface-level. Let the ratio of the depth of the 

 vertex of the one notch below the still-water surface-level to the depth of 

 the vertex of the other be as 1 to n, so that all homologous linear dimensions 

 in the two flows will be likewise as 1 to n. Then, in passing from any cross 

 section of one of the two homologous filaments to the homologous cross section 

 of the other, we have the cross-sectional area x n^, and the velocity of flow 

 oc Vw ; and the volume of water flowing per unit of time, being as the cross- 

 sectional area and the velocity conjointly, will vary as we pass from the one 

 to the other of the pair of homologous filaments, so as to be oc n'Vn. Then, 

 as this holds for every pair of homologous stream-filaments throughout the 

 two flows, if we put Q, to denote the quantity, reckoned voluminally, flowing 

 per unit of time in each of the two entire flows, we have 



Qa Hi. 



Now, as well as considering two separate notches with different streams 

 flowing in them at the same time, we may, when it suits our purpose, con- 

 sider one single notch with streams of different depths flowing at different 

 times ; and if in various cases, either of the same V-notch or of different but 



s 2 



