187G.] ^1^ [Briggs. 



In this equation r = the radius of the opening commencing at the origin 

 of X, and n ^= the radius of the quadrant or corner commencing at the 

 same point. 



The assumption of the quadrant of a circle for the path of the effluent 

 particle from A to B, has been made in order to give a simple equation for, 

 and ready comprehension of, the nature of the sections of the stream nor- 

 mal to its face on A B where, by equalities of areas, uniform velocities 

 would subsist ; but the real curvature (AB) is obviously parabolic, and the 

 plane of B is infinitely distant from (below) the plane of A. Observation 

 has shown, that at about one-fourth D below the plane of A the least sec- 

 tion of vena contracta is apparently reached, and that below this plane of 

 section, the pencil of descending current has its sides with only so much 

 divergence from parallelism, as is due, almost entirely, to the acceleration 

 of the falling stream. An elliptical quadrant which shall approximate to 

 the true parabolic curve can be readily substituted by construction (or 

 calculation) for the quadrant of a circle, in the equation above quoted, and 

 the new values for j, will give loci for the curve of the face of the conoid 

 Z to correspond to the substitution. The value of the radius of the minor 

 axis on the line B as determined by observation, maybe taken as that of (n) 

 in the equation. By this method a very close approximation towards the 

 true form may be attained. ] 



It will then result that the efflux from the peripheral opening C A 

 inwards, having any given velocity, will, in every part of the current, 

 until the least section of the vena contracta on the plane B is reached 

 have a uniform and constant rate of speed ; neither acceleration, nor 

 transformation of head into velocity, will have occurred in the change of 

 direction. If the consideration of the fluid friction, etc., be not taken into 

 the question, and the velocity of efflux at C A is that due to the head, that 

 at B is established and maintained ; whence any liquid particle on the sur- 

 face A B must be in equilibrium of pressure, both from head or momentum 

 in direction of its flow, in loMcJi direction the entire head is transformed 

 into velocity. The plate, or plane surface of D, may be imagined to extend 

 indefinitely in the directions, E E, in which case the velocity of flow of 

 liquid, interposed between the plate E and the bottom A, will decrease as 

 the radial distances from the edge of the aperture; in inverse ratios of the 

 radius r to any new radii r^ r^^ r^,/ : while the height, of liquid column 

 corresponding to the several velocities, r= V at r, V/, at r^, will vary as 



C p. ^ 2 



«&c. The pressure or total height is sup- 



I- 



// 



posed to have been completely transformed into velocity, =r V, at the peri- 

 pheral opening C A, and the stream or sheet of fluid would exert no trans- 

 verse pressure at C A, either upwards or downwards ; Avhile the transverse 

 fluid pressure on the supposed plate or the bottom of the vessel would vary 



(at C A) to —I — —I h '"n being any assumed radial 

 L L r J I rn J -J 



distance from the centre of the opening. Thus if the radius r be taken as 



