ON THE FLOW OF WATER THROUGH ORIFICES. 251 



But work given to it by pressure from behind, while it is passing the 

 initial interface at B, is 



= gp\.c; 



or that work is 



=gm\, 



since pc=m. 



Again, during the emergence of the mass past the interface at U, it gives 

 away to the water in front of it a quantity of work which, in like manner, is 



=p . c 

 = (jpll . c 

 = gmh. 



Also during the passage of the particle from its first place at B to its place 

 at U it descends a vertical space = /; hence during that passage it receives 

 from gravity a quantity of work = gmf. 



On the whole the mass receives an excess of work beyond what it gives, 

 and that excess of work received is 



=gm\-\-cjmf-gmh 

 = gm{7i^+f-h) 

 =gmi; ; 



and as this is the work taken into store as kinetic energy, we have to put it 



= «f. That is, 



or V =>/^' 



which is the result that was to he proved in Theorem I. 



Theorem II. — On the Flow of "Water through Orifices similar in form 



AND SIMILARLY SITUATED RELATIVELY TO THE StILL-WaTER SuRFACE-LeVEL. III 



the flowing of water, from the comUtioii of approximate rest, through orifices 

 similar in form and similarly situated relatively to the still-ivater surface- 

 level* , the stream-lines in the different flows are similar in form: also the 

 velocity of the water at homologoiis places is proportional to the square root of 

 any homologous linear dimension in the different flows : and also (2iressures 

 henig reckoned from the atmospheric pressure as zero) the pressure of the water 

 on homologous small interfaces in the different flows is proportional to the cube of 

 any homologous linear dimension; or, in other words, the fluid pressure (super- 

 atmospheric), per unit of area at homologous places, is proportional to any 

 homologous linear dimension. 



Preparatively for the demonstration of this theorem, it is convenient to 

 establish some dynamic principles, which, for present purposes, may he 

 regarded as lemmas or preparatory pro])03itions, and which will be grouped 

 here together under the single heading of Proposition A. _ 



Proposition A. — If there he two or more vessels containing ivater pent xip in 

 an approximately statical condition, and if they have similar orifices similarly 

 situated relatively to the free level of the statical water — and if we imagine the 



* Or free level of the still water. 



