Lord Rayleigh's Notes on Hydrodynamics. 443 



the case of motion in two dimensions, has been given by Helm- 

 holtz (Phii. Mag. November 1868); and the conclusion that 

 the width of the emergent stream is ultimately one half that of 

 the channel follows from his analysis * . 



This problem throws some light on the formation of a sur- 

 face of discontinuity. If the electrical law of flow held good 

 so that the tube were filled, twice as much momentum as before 

 would have to be generated, and the extra momentum would 

 have its origin in the infinite negative pressure which, accord- 

 ing to that law, must prevail over the extreme edge of the tube. 

 In the absence of forces capable of generating the extra mo- 

 mentum the tube could not flow full. 



A generalization of the problem just considered may be 

 effected by replacing the vessel, whose dimensions were sup- 

 posed to be indefinitely great, by a cylinder of finite section 

 a" (Plate V. fig. 3), in which the fluid moves with finite velo- 

 city v" . If v' and </ be the ultimate velocity and section of the 

 escaping jet, the equation of continuity gives 



t ~t 



g' = v"<j". ....... (4) 



By the principle of energy, 



P = i(v"-v"^i (5) 



and by the principle of momentum, if a- be the area of the 

 tube, 



p ( r = a / v /2 -a // v //2 (6) 



From these equations we obtain 



2 11 



*-? + ?» ( 7 > 



showing that the section of the tube is an harmonic mean be- 

 tween the sections of the cylinder and of the jet. 



The problem of the contracted vein for a hole in a thin plate 

 has been solved mathematically by KirchhofTf for the case of 

 motion in two dimensions. As this solution is very little 

 known, and many points of interest are passed over by Kirch- 

 hoff himself, a short account of it accompanied by a few 

 remarks and calculations may not be out of place. 



With the notation explained in the previous paper, the form 

 of f proper to this problem is 



5 = e-»+ >/«-*» — 1 (8) 



* The application of the principle of momentum to the case of the in- 

 troverted tube was original with myself, but, as I learned at Glasgow, had 

 been made previously by Mr. Fronde. In small-scale experiments the 

 result is liable to be vitiated by adhesion to the sides of the tube. 



t Crelle, vol. lxx. 1868, and Vorlesungen iiber mathematische Phijsik. 



