ISiS."] 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



245 



The e.xperiments of Michelotti give for the diameter of the 

 contracted column nearly the same results. The distance of the 

 greatest contraction from the orifice is estimated by him smaller, 

 so that it more nearly agrees with the radius of greatest contrac- 

 tion, than with the radius of the orifice. But it appears that the 

 exact measurement of these distances is subject to various difficul- 

 ties, and that the difference lies at least not wholly beyond the 

 limits of the uncertainty of measurement. 



7. Remarks and Inferences. 



1. The contraction, according to the above investigation, arises 

 from the sudden change of direction and velocity to which the 

 particles of water immediately in the plane of the orifice are sub- 

 jected. It depends (so far as the above experiments leave to be 

 safely inferred) only on the radius of the orifice; whence it fol- 

 lows, that the force of contraction is proportional to the radius of 

 the orifice. 



2. For circular orifices, all the diametrical contractions will be 

 equal and opposite; whence it follows, that the sections of the con- 

 tracted stream will be similar. 



3. When, on the other hand, the different diameters of the ori- 

 fice are unequal, the sections of the stream vary in form, while the 

 distance of the greatest contraction from the orifice will (1) be 

 jiroportional to the several diameters or secant lines, and therefore 

 will not be equally distant from the orifice, nor be in one plane; 

 which must be considered as the condition of the similar form of 

 section of the stream. 



4. The above theory of contraction assumes that the reservoir is 

 large, the movement of the water free, and the orifice completely 

 isolated from the bottom and sides; also that the altitude of pres- 

 sure is so large, that the depression of tlie surface which takes 

 place above the orifice is inconsiderable. Without tliese condi- 

 tions, the regular effect of the contraction would be intermitted; 

 and in such cases, since the law of the irregularities is not yet 

 known, we must for the present be content with an approximate 

 computation. 



5. The greatest mean velocity V determines the distance of pro- 

 jection of the stream, and takes place in the point of greatest con- 

 traction: we have, therefore, 



(■» ^' = (i.'Jc^ = jFr 



since Q = C. m (§5). 



6. The accurate determination of tlie quantity of discharge from 

 the orifice, depends on the correct determination of the velocity V. 

 Usually this is obtained by means of the altitude of pressure above 

 the centre of the orifice, by the formula V ;= VC-l^H), which ne- 

 glects the influence of the height and figure of the orifice. This 

 influence is considerable, but is smaller as the altitude of pressure 

 increases, and for great altitudes may therefore be safely neglected. 

 For a general investigation of the question, a more accurate and 

 general computation of the velocity is however necessary. 



7. The friction of the sides of the orifice, the difference between 

 the pressure of the air at the surface of the water in the reservoir 

 and the orifice, the resistance of the air against the issuing stream, 

 and tlie influence of temperature on the quantity of discharge, 

 must for the present remain unconsidered, as the foregoing experi- 

 ments are insuflicient to determine these small various effects. 



The mean velocity of water discharged through orifices in thin plates. 



In the tlieorem of Torricelli, the altitudes of pressure are the 

 abscibSBP, and the velocities generated by them the ordinates of a 



parabola of which the parameter = ig. If, therefore, in the pa- 

 rabola (fig. 2), AB be the axis of abscissae, z- = p.r. 



In the theorem of Torricelli, V- = ig)i ; and, therefore, 

 3 = V; /) = ig; and j; = H. 



If the orifice be made in the bottom of a reservoir, and its sur- 

 face be horizontal, the depth of water above it, or its altitude of 

 pressure, is equal for every point in the orifice. Let, therefore, H 

 designate the altitude of pressure; the velocity of issue for a hori- 

 zontal orifice is found directly from the above equation — that is, 

 V = V(%H). 



If, on the other hand, the orifice be vertical, as in one of the 

 sides of the vessel, every horizontal section of the orifice has a 

 different depth below tlie surface of the water; and then the mean 

 velocity of issue for all the different velocities in the vertical ex- 

 tension of tlie orifice has to be calculated. 



If ACF (fig. 2) be the vertical plane through the centre of the 

 orifice, eV = v the diameter of height, Ae=H the altitude of 

 pressure at the upper edge, the velocity for the altitude Ae is 

 equal to erf, and the velocity for the altitude AF (at the lower 

 edge of the orifice) = Fe. The mean velocity between e and F, 

 in the vertical section of the orifice, is therefore the mean value of 

 the ordinates of the parabola between the limits ed and Fc, whicli 

 may be easily found, as^ in §2, — namely, 



J'zdx V(%)jA-* 



fd. 



since z- ^= ig^- 



Taking this integral between the limits x 



dx 



fdx 



we find 



II, and .r = II -f c, 



(E) 



. = v = IW%){^^)}--^}. 



This expression gives the mean velocity in the vertical plane of 

 the orifice eF, under the altitude of pressure H. 



9. 



A column of water may be considered as made up of an indefi- 

 nite number of slices parallel to the vertical section; and every 

 slice again may be represented as a very large number of threads 

 of water. Give, now, to each thread a length equal to its velo- 

 city: and so a prismatic body is obtained of which the mean lengtli 

 or height gives the required velocity. 



To estimate this length more closely, let the origin of co-ordi- 

 nates be transferred to e (fig. 2): for the length of every thread of 

 water, or what is the same thing, for its velocity, we ha\e the equa- 

 tion Z' = ig (H -|- ,r). 



Let y designate the indeterminate ordinate of the orifice, of 



which the area is therefore = I ydx, and the content of tiie 



prismatic body z / ydx, or rather = / zydx, since z is variable 



and a function of ,r. To find the mean length or height of this 

 prism, we must divide its content by the sum of the threads of 



water, or by the area of the orifice — that is, by / ydx. In this 



manner we obtain generally — 



/ zydx 

 (F) ~=V = -^^ ; 



Substitute for z the above value, and we have — 

 V{ig)J'ydx^{n^x) 



(G) V = 



/' 



ydx 



And this is the expression for the mean velocity with which 

 the water at any vertical orifice whatever in a thin plate is dis- 

 charged. 



The velocity being known, the quantity of discharges at the 

 orifice is easily determined. Let the quantity of discharge = tj, 



