and on the Maximum Effect of Machines. 419 



1 "From these results it appears, that when the contracted vein 

 is not opaque, and when its velocity is nearly equal to that 

 which is due to the head, the reaction is nearly equal to what 

 it was concluded to be by Sir Isaac Newton and M. Dan. Ber- 

 noulli; and the great apparent difference between Sir Isaac 

 Newton's first and second conclusions arises from his having 

 been misled by some experiments to which he alludes. He 

 says — " Per experimenta vero constat, quod quantitas aquae, 

 quae, per foramen circulare in fundo vasis factum, dato tem- 

 pore effluit, ea sit, quae cum velocitate praedicta," [viz. the velo- 

 city due to the head] " non per foramen illud, sed per foramen 

 circulare, cujus diametrum est ad diametrum foraminis illius 

 ut 21 ad 25, eodem tempore effluere debet*." We must pre- 

 sume, however, that he refers to experiments made by others ; 

 for if he had made them himself, he would, no doubt, have 

 arrived at the same results which have since been so well esta- 

 blished by various authors, and he would have stated the above 

 ratio to be as 19*5 to 25 nearly. 



But his demonstration of the reaction requires that the ve- 

 locity at the contracted vein shall be equal to that which is 

 due to the head. Now that velocity cannot be determined by 

 measuring the imperfectly contracted vein in cases of water 

 spouting through a hole in a thin plate. 



We may safely indeed infer, that, in such cases, the velo- 

 city is considerably less than what is due to the head. For, 

 the jet being opaque, some moving force must be expended in 

 separating the particles from each other, and the distance to 

 which the jet from such an orifice is projected on a horizontal 

 plane, confirms that inference. The demonstration, therefore, 

 of the reaction, can be properly applied to such cases only as 

 those where the water, issuing through a tube properly con- 

 tracted, acquires the velocity nearly which is due to the head, 

 and in those cases the experimental results agree, as I have 

 stated, remarkably well with the demonstration. 



These results agree also with the explanations which have 

 been given of moving force f . If we suppose the velocity of 

 the jet to be equal to that which is due to the head, and the 

 vessel to move uniformly in the opposite direction CD with 

 the same velocity; the water will be at rest as it issues. 



Let a represent the area of the smallest section of the ori- 

 fice. Then while the vessel has moved through a space as 



2 BC, a quantity of water represented by a x 2BC has de- 



* Principia, edit. ii. lib. 2. prop. 36. 



t By moving force is meant the product of the pressure into the space 

 through which it acts, or of the quantity of water into the height through 

 which it falls. The same sense in which the term is used by Euler. 



3 H 2 scended 



