ON THE FLOW OF WATER THROUGH ORIFICES, 253 



Hence the centrifugul forces are 



as 



r nr 



or as 1 : n^. 



This being understood, it readily becomes evident that if, instead of small 

 solid masses sliding along guides, we have two small homologous masses of 

 water m and m, fig. 7, flowing in similar slender guide-tubes, and if homo- 



Fiff. G. Fig. 7 



logons pressures be applied to the two masses in front and behind which 

 are as 1 ton', and i^at homologous situations in their two paths their 

 velocities be as 1 to Vn, then, in respect to all the forces received by the two 

 masses from without, other than those applied by the guide-tubes, and also 

 in respect to the forces required to be received for counteracting their centri- 

 fugal forces, we see that aU these constitute force systems similar ^^ 

 arrangement and of amounts as 1 to n\ It therefore follows that the 

 forces which the masses must receive from their guide-tubes must be similarly 

 arranged and of amounts, on homologous small areas, as 1 to n\ 



This being settled, wo may now pass to the demonstration of Proposition 

 A, at present in question. . 



Suppose No. 1 and No. 2 in fig. 8 to represent two similar vessels with 

 similarly guided flows, in all respects as described in the enunciation of this 

 proposition. Let W L and W L' be the stiU-water surface-levels, or the free 

 levels of the still water in the two cases. Let B C D, B' C D' be two similar 

 bounding interfaces, each separating the region of flow with important energy 

 of motion from the region which may be regarded as statical, or as devoid 

 of important energy of motion. Let B U E in No. 1 and B U E in No. 2 

 be two homologous guide-tubes, and let them for the present be understood as 

 terminating- at two homologous cross interfaces E and E', which may con- 

 veniently be understood as being each situated at a moderate distance outside 

 of the orifice— for instance, at some such place as that which is usuaUy 

 spoken of as being the " vena contractu," or where the water has attained a 

 pressure not diff-ering much from that of the atmosphere, or it may in some 

 cases even be that the atmospheric pressure is there attained ; but the exact 

 places at which to suppose the homologous terminations E and E of the two 

 guide-tubes as being taken are not at all essential to the demonstration. 



Let homologftus linear dimensions in No. 1 and No. 2 be as 1 to n. 



Let the velocity at any variable point U in the guide-tube B U E be denoted 



^Let the pressure at U, expressed in units of pressure-height, be denoted 

 by Ti ; as shown by the vertical line U T in No. 1, where T is the top of the 

 pressure-column for the point U. 



