354 CALCULATIONS for 



you pleafe parallel to the horizontal line IG. Now if we take 

 the ordinate AI, or its equal DH to exprefs the whole uniform 

 velocity acquired by falling from C to D, it is evident that the 

 ordinate OP will denote the velocity at the point O, acquired by 

 a fall equal to CO, and the ordinate NK will exprefs the veloci- 

 ty arifing from a fall equal to NB or QS. But we ftiall prove 

 that the velocity of the afcending water in the fecond branch, 

 when it arrives at Qj, ought not to be expreffed by the ordinate 

 which correfponds to it ; but by the line LK, the difference be- 

 tween the entire uniform velocity LN or MB (by falling from A 

 to B) and that of NK. 



To demonftrate that the height QS or NB of the water in the 

 tube SR, is equal to a fall which can produce the relative velo- 

 city arifmg from CD, or the difference between the velocities 

 acquired by falling from A to B, and that of the afcending 

 water at Qj let AB and QS be confidered as two non-elaftic 

 bodies, whofe momenta are as the altitudes AB and QS. If 

 AB = a and QS = r, we fhall have a — VaxVa, and r = Vrx 

 ^/r ; but the difference of the momenta divided by the fum of 

 the bodies is equal to the velocity, which let be v ; therefore 



4/^X^^— ^rx-/rdividedby^/^+^/^= '^^^'^^'^^^f~^' ^ = 



Va-\-vr 



4^/a — v^r = v= LK, the velocity of the water at Q^, and 'Z^— 



V = v'r, which is the relative velocity produced by a fall equal 



to QS. As this velocity is expreffed by the ordinate NK, the 



difference between it and MB or LN will exprefs the retarded 



velocity of the water in the tube of communication DX, which 



is the fame as that of the furface QR at the point Q^ 



As it will be the fame with all the retarded velocities during 

 the time employed in filling the tube GF, it follows that their 

 fum will be expreffed by that of all the ordinates, or the area 

 of the parabolic complement MIKB. 



Before the obfervations of Belidor on the inverted fyphon, in 

 his theory relating to the common fucking pump, it was cuf- 

 tomary to eftimatc this fum by the area of the parabola DCPH 

 or ABKI ; for the velocity at Q was expreffed by the fquare root 

 of CO, inflead of the difference between the fquare roots of 

 CD and QS. 



The parabolic complement MIKB, being but half of the pa- 

 rabola ABKI, it is evident that the fum of all the retarded 



velocities 



