Prof. Magnus's Hydraulic Researches. 181 



Explanation. 



65. The parts of the liquid arrive at the orifice in very differ- 

 ent ways. The motion of each particle, and we will first consider 

 that of a particle at the edge of the aperture, can be divided into 

 a vertical and a horizontal component, and this latter can be 

 again divided into two directions, of which the one is normal to 

 the perimeter of the aperture, and the other at right angles to 

 the normal. 



It is of coui'se impossible to determine the magnitude of the 

 motion of such a particle in difi'erent directions during the efiiux, 

 as also the relation of these motions for the whole of the particles 

 which simultaneously pass over the edge of the aperture, since 

 they may have attained the edge in very difi'erent ways. But 

 before the liquid begins to issue, the whole of the particles at 

 the bottom are under the same pressure. All those at the edge 

 of the aperture move, when the latter has been opened, with equal 

 velocity over the edge, since it is assumed that up to then no 

 motion has occurred in the vessel. Hence they all begin their 

 paths in the direction of the normal with equal velocity. But 

 having passed over the edge of the aperture, there is in addition 

 to the horizontal, a vertical motion. The resultants of these two 

 change their direction continually, partly in consequence of the 

 gravity of the particle, and partly in consequence of the pres- 

 sure which the mass of liquid above exercises. But a change in 

 the direction of the motion is principally effected by the cohesion 

 which takes place between the parts of the liquid, and by the 

 resistance which the mass of water in the interior of the jet 

 0])poses. These last actions of cohesion and of resistance induce 

 principally the peculiar forms of the jets, for they are not the 

 same for all particles passing from the edge of the aperture. 



66. Let us consider first the resistance of the liquid in the 

 interior of the jet to the horizontal part of the motion of a par- 

 ticle passing from the edge of the orifice to the cross section of 

 the jet immediately under it. It is manifest that if the orifice 

 were a circle, this resistance would be equal for all particles 

 which came from the circumference of the aperture. But if the 

 aperture be not a circle, and those radii of curvature of its peri- 

 meter which fall in the aperture be called positive, and those 

 without negative, then the resistance to a force acting from 

 without must be less at those points whose radii of curvature 

 arc positive, than at those points at which they are negative ; 

 and it will be the less the smaller the positive radius of curva- 

 ture. For at that part of the perimeter at which the radius of 

 curvature is positive and smaller than at the adjacent parts, 

 which therefore project more, the particles of the liquid are more 



