490 



HYDRODYNAMICS. 



from 

 Orifices. 



Discharge may be supposed to be divided, and therefore the sum 

 of Fluids of a ll these elementary terms, which may be obtained 

 either by fluxions or by a geometrical construction, will 

 be the time required in the proposition. 



In order to find the time geometrically, draw EF 

 equal and parallel to BC, and construct upon EF as an 

 axis a parabola FTG, with a given parameter p. Prolong 

 the lines AB, MN, ft v, and US, till they meet the para- 

 bola in G, c, d, and T. Construct a second curve XZ Y, 

 so that each of the ordinates Ha, K ft, LZ, may be equal 

 to the corresponding sections MN, p >, RS, divided by 

 their corresponding ordinates in the parabola H c, K d, 



MN 



LT. Now, since H a =-p , and MN = H a X H c, 

 H c 



and since by the property of the parabola, ( See CO- 

 NIC SECTIONS, Prop, XIII. p. 157.) H c* = HF x p, 



and y'HF r= \/P m = 7 we have, by substituting, 



in the above value of /, the preceding values of MN, 



HoyHcxHKx Vp i , -r 

 and ^P ,n, t = 2Ax ^ gxtic ' and dlvuhn S 



fay </p and H c 



which consists of the constant factor --7- multi- 



2 Ay'g 



plied into the variable curvilineal area H a b K. But 

 as the same may be shewn for every other element of the 

 time, it follows that the time of descent from AB to 



RS will be equal 





X ELZX. 



/m o Discharge 



. But in uniform motions, the spaces divided by of Fluids 



g from 



the velocities are as the times of description. Hence Oiificn. 



2 mo HK /mo _. - IC * . . , ^~"" Y ~" < ' 



-=r=r : -rr- = / : Time HK, (or the time of de- PLATE 



EG He V g ,wvv 



scribing GV.) Consequently 

 TimeHK = 



CCCXV1IK 



EG 



and substituting for /m o its value . , p being the 



parameter of the parabola, we have 

 IP 



Cor. It follows from this proposition, that the times 

 in which the surface AB will descend through the 

 heights o P, o s, will be proportional to the correspond- 

 ing areas EH a X, ELZX, and that the time of descent 

 through any of these heights is to the time in which the 

 vessel is completely emptied, as the corresponding area 

 EH a X or ELZX, is to the whole area EFYX. 



PROP. VII. 



To determine the time in which the surface of water 

 in a prismatic or cylindrical vessel will descend through 

 a given height, viz. from AB to RS in Fig. S. 



This problem, as Bossut has remarked, may be very 

 easily resolved by the method of fluxions ; but we shall 

 follow this excellent mathematician in the elementary 

 demonstration which he has given of it. Let us sup- 

 pose that a body, not heavy, ascends through the height 

 FIATE "* > ^'6- 3- an d describes that space in the very same 

 eccxvm wa y ^ a heavy body would descend through the height 

 Fig. 3. ' om. Then it is obvious that the different velocities of 

 the ascending and descending body may be expressed 

 by the ordinates of a parabola GTF. When the ascend- 

 ing body has arrived in *, it will describe the small space 

 x P or KH, with a velocity represented by the ordinate 



H c ; but the time^of describing m o is J ; and if the 



8 



final velocity of the ascending body were continued uni- 

 form, the body would describe a space=2 m o in the time 



but by Prop. VI. the time in which the water descends 

 through the same space P -, or HK, is 



ELxHaxflK-^-X^. 



MN 

 If we now substitute in place of H a its equal ~ ; and 



i 1 C 



multiply the first of these expressions by MN, and the 

 second by A, the products will be equal, or 

 MN.y/p.HK_ MN . A >/p . HK 



2 vVr He 2 A \/g .He 



Hence, by Euclid, (VI. 16.) the time of the body's as- 

 cending through m o, is to the time in which the sur- 

 face descends through P v, as the area A of the orifice 

 is to the area MN of the base of the cylindrical or pris- 

 matic vessel ; and as the same is true of all the other 

 elementary times which the ascending body and the de- 

 scending surface employ in describing small equal 

 spaces, it follows that the whole time in which the as- 

 cending body will describe the height m o, is to the 

 time in which the vessel will be completely emptied, as 

 the area A of the orifice is to the area of the base of 

 the vessel. The time, therefore, in which the vessel 



will empty itself will be ^/ X -j-, B being the] area 



of the base. 



If RDSC is the vessel, then the time in which it will 



be entirely emptied will be ^J X -r-> consequent- 

 ly the differences of these times, or the time in which 

 the surface AB will descend into the position RS, will 

 be 



PROP. VIII. 



To construct a clepsydra, or water clock, of a cyliu- 

 drical form. 



The equation in the preceding proposition enables us Fig. ". 

 to do this in a very simple manner. Let us suppose that 

 it is required to measure 12 hours, and that the height 

 AD is divided into 144 equal parts ; then the height of 

 the surface of the water at the commencement of the 

 time will be 144 parts. At the end of one hour the 

 height will be 121 ; at the end of the second hour it will 

 be 100, as in the following Table : 



Hours to run 12 11 10 9 8 7 6 5 4 3 2 1 



Hours from commencement 01 234567 8 9 10 11 12 



Height of the surface from the bottom 144 121 100 81 64 49 36 25 16 9 * 1 



Length of each hour in parts 23 21 19 17 15 13 11 9 753 1 



