NEWTON S PRINCIPJA. 259 



efflux of water, as to its velocity, will remain the same as 

 before. It will not be less, because the ice now dissolved 

 will endeavour to descend ; it will not be greater, because 

 the ice now water cannot descend without hindering that 

 of other water equal to its own descent. The same force 

 ought always to generate the same velocity in the effluent 

 water. Newton does not mean that the same quantity of 

 water flows out, but that the same velocity is generated. 

 This is not, however, the vertical velocity at the orifice ; 

 &quot; for the particles of water do not all of them pass through 

 the hole perpendicularly, but flowing down on all parts 

 from the sides of the vessel and converging towards the 

 hole, pass through it with oblique motions, and in tending 

 downwards meet in a stream whose diameter is a little 

 smaller below the hole than at the hole itself.&quot; The 

 particles of water at the actual orifice are not all moving 

 in the same direction. Those in the centre are descending 

 vertically while those near the circumference have a lateral 

 motion, but at the vena contracta the whole fluid is 

 descending vertically. If therefore B, B be the areas of 

 the sections of the vena contracta and orifice, and v the 

 velocity of the fluid at the vena contracta, the mean velocity 

 of the fluid perpendicular to the orifice will be clearly 



B 



& 9 



The quantity of water that flows past any horizontal plane 

 is proportional to the product of the area by the mean 

 perpendicular velocity. Thus the discharge is equal to 



q = B V2g(h + k) 



where h is the depth of the vena contracta below the sur 

 face of the fluid in the vessel. 



T5 -I 



Newton found the ratio ^= - nearly, whence the 



S 2 



