1870.] '^1^ [Briggs. 



once become evident, that there are two normal forms for the veyia con- 

 tracta, viz.: That, when the stream emerges downwards from an opening 

 through horizontal edges, and that, where it emerges upicai'ds through an 

 opening of the same character. The first of these gives a pencil, whose shape 

 for its longitudinal section at its upper end, or origin, will be controlled by 

 tlie nature of the aperture, and by the eflFect of the initial directions of 

 the particles of the effluent liquid (the theoretical vena contractu, under 

 pressure, btit devoid of gravity); modified by the effect of gravity, which 

 would give to any descending pencil of a fluid, the motion of whose par- 

 ticles shall be established in approximately parallel lines, a hyperloid con- 

 tour. The second of these will give a sheaf, whose shape at the point of 

 efflux, will be determined by the same laws ; while it would now be modi- 

 fied at this point, by the load of the emerging fountain, and at the same 

 time the form of tlie stream above (in this case attaining on some plane an 

 absolutely contracted section), would be that of a hyperboloid sheaf, with 

 both external and internal lines of definition. If it be supposed in this sec- 

 ond instance that the plane of efflux (of the orifice) is slightly deviated 

 from the horizontal, so that the emerging stream is made to take a line out 

 of the perpendicular one, the sheaf form would be disturbed; and at some 

 quite small angle of deviation, a trajectory curve would take its place. 



The general course of the stream would then have a modified parabolic 

 curvature — a trajectory curve, Avhich has been frequently discussed — but the 

 exact contour of the pencil is still an open question. It is certain than 

 when passing the point of greatest elevation, it would have, from its re- 

 tarded motion, its greatest cross-section, and that this cross-section would 

 be a flat oval of peculiar form ; and it is probable that beyond this section, on 

 the descending stream, it would become nodal, for the same reasons that 

 a stream emerging from any orifice except a circular one becomes nodal.* 

 In short the complete solution of the problem not only admits and assumes 

 values for all the physical conditions, but it will embrace all directions of 

 efflux from 0° to 180°, where 0° may be taken as the perpendicular direc- 

 tion, either upwards or downwards. 



It is possible, for the purpose of illustration, to give some consideration 

 of the vena contracta upon hypotheses similar to those of Professor Thom- 

 son, and if other conditions are assumed at the same time, an appreciation 

 of the phenomenon can be had. In truth the view it is proposed to offer 

 may go further than a mere appreciation, and may be made the basis for 

 support of the other fundamental controlling conditions, and indicate the 

 true line of procedure for mathematical investigation. Let us suppose, 

 with Professor Thomson, that the effect of fluid friction, or viscosity, is 

 neglected; that the magnitude of the vessel and the depth of liquid, is so 

 large in relation to the dimensions of the orifice, that no appreciable velo- 

 city is imparted to the mass of liquid by the discharge; that the jet is one 

 issuing downwards (so as to have the cross-section under absolutely uni- 



* .See article by Weisbach, " Ausfluss," In the " Allgemeine Maschinen Eucy- 

 clopadie." 



