ON THE FLOW OF WATER THROUGH ORIFICES. 255 



Hence putting v, and {v^)' to denote the velocities at E and E' respectively, 

 we have (by Theorem 1., which proves that the velocities must be proportional 

 to the square roots of the falls of free level) 



v,:{v,y: : vT: Vh, 



«r {vj = v,'>/n (1) 



Again, from similarity of forms, we have iu respect to areas of cross- 

 sections of the two guide-tubes : — 



area at E area at E' 

 area at U area at U' ' 



or since reciprocals of equals are equal : — 



velocity at E velocity at E' 

 velocity at U ~ velocity at U' 



V V 



or by ( 1) — '=-£-^, 



or v' = vsln. . (2) 



This applies to any or all homologous points in the two regions of flow. 



Now by referring to the figure or otherwise, it will readily be seen tliat 

 I, or the fall of free level from B to U, is =li^-\-f—h, while ic=:\ — h; and 

 that therefore ^=f-\-'k. Hence, by Theorem I., we have 



v= ^2g(f+Jc). 

 In like manner in No. 2 : — 



t;'=V%Cf+F); 

 but by (2) v'=»Jn.v. 



Hence V2^(f+W)=. ^n . V2^f^F), 



or f'+k'=nf+nk. 



But by similarity of forms f'=nf. 



Hence, subtracting equals from equals, we have 



k' = nk; (3) 



but by similarity of forms {h^)'=nhi (4) 



Also, since the pressure at any point in a stream-line, or guide-tube, is its 

 initial pressure minus the relief of pressure, we have 



h^-h=h (5) 



and (h^)'-k' = h' (6) 



