44(i Lord KayleiglTs Notes on Hydrodynamics. 



A few pairs of corresponding- values of y and v will give an 

 idea of the relation expressed in (17). 



' 10 



y= -1-834 



v=i 



y= — *613 



v=% 



y=- -1095 



V =1Z 



y=+ '3792 



v=i 



y=+ -902 



v=i 



y= +3-489. 



By interpolation we find that, corresponding to y = 0, ^ = '6840, 

 r 2 = *420. Hence the pressure in the middle of the orifice is 

 '58 of that prevailing in the vessel, the external pressure being 

 treated as zero. In these statements the ultimate velocity is 

 understood to be unity, and the scale of linear magnitude is 

 such that 2 + 7r represents the width of the orifice. 



Meeting Streams. 



The principle of momentum gives interesting information on 

 the question of the mutual action of streams which come into 

 collision. Suppose, for example, that the motion is in two 

 dimensions, and that two equal streams moving with the same 

 velocity meet at an angle 2« (fig 4). After the collision the fluid 

 resolves itself into two other streams of unequal width parallel 

 to the line bisecting the angle 2a ; and a question arises as to 

 the relative magnitude of these streams. The ultimate velocity 

 is of course in both parts the same as before the collision. 



The width of the original streams being unity, let us sup- 

 pose that the width of that derived stream which is least 

 diverted is x. The width of the other derived stream is then 

 2 — x ; and the principle of conservation of momentum gives at 

 once the relation 2 cos u = x-(2-x), 

 whence x=l+ cos a. The ratio of the two derived streams is 

 2-^:^=tan 2 ^ (18) 



For example, if a = 60°, the ratio of the derived streams is 

 1:3. The effect of friction would be to make the ratio still 

 more extreme. 



If we suppose the motion reversed, we obtain the solution 

 of the problem of the direct impact of two streams of unequal 

 widths which meet with equal velocities. The ratio of streams 

 being known, (18) determines the angle of divergence. 



In the case where a. is a right angle the four streams are all 

 equal, and the bounding surfaces are symmetrical with respect 

 to the straight lines bisecting the streams (fig. 5). The exact so- 

 lution of this case has been indicated by Kirchhoff. If 6 be the 

 angle between the tangent to the free surface at any point and 



