344 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[August, 



orifice is at such a distance from the side-plates and bottom, that 

 little disturbinjj influence exists. 



.'\1I the particles of water in the horizontal plane AFB sustain 

 the same pressure, since this plane is parallel with tlie surface of 

 the water in the reservoir. The direction of the motion of the 

 issuing particles heinjjf towards the centre of the openinj;, it follows 

 that the velocity of all points equally distant from the centre must 

 be ecjual, because the pressure with which the motion i)roceeds is 

 on all sides equal; or, in other words, all the particles in the cir- 

 cumference of the semicircle AFB must have the same velocity 

 towards the centre of the orifice. 



Let the semicircle .'VFB be turned about its axis F;h until it be 

 in the vertical plane: then the pressure above Fm is smaller, and 

 beneath greater, than in Fm itself. Call H the altitude of pres- 

 sure at the centre of the orifice, or in Fm; +a the distance from 

 Fm for a lower, and —a the distance for a higher point: then 

 5 (H -ha -I- H — a) = H ; and since Fm bisects the semicircle, the 

 mean altitude of pressure in all points of the semicircle turned 

 through an angle of 90° or vertical, in like manner = H. This 

 remark holds good when the semicircle is turned through any other 

 angle than 90^'. Let, therefore, the semicircle make a complete 

 revolution al)out its axis, so as to describe the surface of the hemi- 

 sphere of which the centre is the centre of the aperture: the mean 

 value of the pressure which tlie particles of water in this surface 

 sustain, for the above reason, is the same which a point in the axis 

 Fm sustains — that is, = H. 



Now, since the mean pressure of all the particles in the surface 

 of the hemisphere is equal to the pressure which at the centre of 

 the opening takes place in the direction of c, the mean motion of 

 all those particles will be equal to that which takes place in that 

 direction (provided that the opening be not very large); whence it 

 follows that all the particles in the surface of hemisphere AFB 

 have the same velocity. 



The surfaces of the hemispheres diminish towards the orifice in 

 proportion to the square of their radii; the velocities must there- 

 fore be in the inverse proportion which the surfaces follow. Con- 

 sequently, as was above stated, the velocities of the concentric 

 shells of water are in the inverse proportion of the squai-es of their 

 distances from the orifice; and if e = Fc, and e =/'c, we have, as 

 above, 



(Fc)-.v = (fcy.v. 



The motion proceeds in this relation until the radius of the last 

 shell of water is equal to the radius of the orifice : then occurs an 

 alteration in the direction as well as the velocity. The particles 

 which proceed from B to b towards the centre c, at the instant of 

 reaching the orifice are acted on by pressure in the direction bk, 

 jiarallel with cf. They possess also already a certain velocity to- 

 wards the centre c, aiul" move therefore in the direction bp. Another 

 thread of water moving from G to g, is acted upon by pressure in 

 tlie direction gi ; and so on. In this manner, at the orifice, all the 

 particles in the surface of the hemisphere are suddenly acted upon 

 in a direction parallel to cf, and alter their velocity together; up 

 to the particle of water in/ which moves along the axis itself, and 

 as neither its direction nor velocity is altered, moves with the great- 

 est velocity, which according to Torricelli's theorem = /^{'\-gll). 

 Let the velocity which the altitude of pressure H produces be de- 

 signated by V; hence, V = V(*tfH). V is the greatest velocity 

 of discharge, and exceeds the velocity which takes place in the 

 plane of the aperture by a certain factor which is called the co- 

 efficient of contraction. Let the mean velocity in the orifice be C, 

 which is also the mean value of all the velocities with which all the 

 particles of water pass the orifice. In order to find this value, we 

 must ascertain, for the equal velocity of all the particles, their 

 mean distance from the axis ab in all the sections passing through 

 the axis. But the particles in the periphery of the circle a/v, and 

 in the surface of the generated hemisphere, have the same velo- 

 city: their distance in this periphery from tlie orifice will be there- 

 fore the perpendicular, as gi, — or what is the same thing;, it will be 

 the ordinate y in the equation to the circle y- =; 2rjc—x^, in which 

 lib is the axis of abscissse. This is the case for every position of 

 the semicircle turned about its axis. 



The sum of all these ordinates is the area of the semicircle a/v, 



or = jydx. Now, the mean value of y, which may be called y , 

 may be put as a function of the same form and value, in which y is 

 invariable, and consequently we have for it y I dx; and when the 



two expressiens are equated, we have y j dx =■ lydx; and, con- 



Jydx 

 sequently, y = . 



fdx 



This integral between the limits x = 0, and x = 2r, gives 



y — IriT; 

 and thence, according to the proportion (A), 

 (B) c : \ = {\r.Y : r"- ; 

 or, (C) c = {\^f = (1^)2 V(tflH) = O-r.17 VC^t/H). 



The co-efliicient A' is therefore (i")" = 0'617 



The experiments of Bossut give this co-efficient = 0'617 



Evtelwein makes it := 0-6176 



D'Aubuisson := 0'617 



Other experiments =: 0'619 



The agreement of theory with experiment is therefore so complete 

 as to leave nothing to be desired. 



3. 



For the velocity perpendicular to the axis c/'(fig. 1), the dis- 

 tance i?« above referred to is found; and if z designate the corre- 

 sponding velocity, we have by the proportion (A) 



z:c = (gsy : {giy = 1 - (.^)= : Q^f ; 

 or, .= c{ay-l}. 



And when these velocities are estimated by the Parallelogram of 

 Forces, we find for their direction 



t^"T = ^= 0" -^' 



which gives y = 31° 51' 6". Hence it follows that for the angle 

 which the tangent of the issuing column makes with the radius of 

 the orifice, 



90° — 7 = 58° 8' 54.". 



Poncelet and Lesbros (^^Experiences hydrau/ique siir les his de 

 recoulement de I'eua.' Paris, 1832; Table 5) have, by their experi- 

 ments at Metz, made, propably, a very accurate measurement of 

 the column issuing from a square orifice, and, as far as the drawing 

 indicates, find very nearly the same angle; which besides, as we 

 shall see further on, by the co-efficient, is, for the square orifice, 

 not quite equal to, but on the average must be found something 

 smaller than, that for a circular orifice. 



4. 



We have seen that the mean velocity with which the particles 

 pass the orifice is less than V. But as in the plane of the orifice 

 itself, all sustain the same pressure, H, which produces the velocity 

 V, they must be accelerated outside the orifice — and, indeed, up to 

 a point where they obtain their velocity which belongs to the alti- 

 tude of pressure H. Let this distance, therefore, be x: it follows, 

 by the proportion (B), 



C : V = {Ir.y : ,t"; 



and when for C its value in § 2 is put, we find ,r = r : that is, all 

 the particles obtain outside the orifice the greatest velocity V, first 

 at the distance r from the orifice; whence it follows, tfint the point 

 of greatest contraction of the column of icater takes place at a distance 

 from the opening equal to its semi-diameter. 



5. 



In the proportion (B) § 2, (^^rir)- : r- is also the proportion of 

 the normal sections of the column of water; and thence it follows, 

 that when r is the radius of the orifice, -^jrit := p is tlie radius of the 

 column at the jioint of greatest contraction. 



For the quantity of water issuing through the different sections 

 of the opening is always the same; aud if the sections correspond- 

 ing to the velocities c and V be designated a? and Wj, we have 



a> , c 



Q = wc = ""jV; and, therefore, Wj = -vr-- 

 But til =: (.^n-; c = (j'')'V; consequently. 



And, therefore, p = ^rir = r.0-785i. 



Poleni's experiments ('iViwivi raccolta d' autori che trattano del 

 vioto dell' acque' — delle pascnje 111.; Parma, 176()) give p= r. 0-7884; 

 and Borda {^Mem. de Paris,' 17(j6) found by direct measurement, 

 !>= r . 0-S02. 



