70 EEPOBT--1881. 



stream line, c describes a broken path, viz., a carve, which corresponds 

 to the rigid walls ; and part of a circle, which corresponds to the free 

 surface. For the region of to are taken two constant values of \p, and ^ 

 varying between — oo and 4- oo. The boundary of c chosen is a crescent, 

 one of whose arcs has its centre at the origin, and is to correspond to 

 the free surfaces. Marking on this the points which correspond to the 

 infinite branches of the walls and free surfaces respcctivel}', the points 

 of the crescent will correspond to the ends of the walls where e changes 

 from a given direction to a given magnitude. The problem now is to 

 find the relation between e and w, that the regions of c, w may be 

 transformed into one another, so as to be similar in their corresponding 

 small elements. Several extremely interesting problems are solved in 

 these two papers.' 



In one case only KirchhoS" determined the pressure exerted per imit of 

 length of an infinitely long plane strip immersed perpendicular to a 

 stream, but this is the only practical application he makes of his results. 

 Lord Rayleigh^ has deduced several most interesting conclusions by 

 means of Kirchhoif's formulas. He finds expressions for the moan 

 pressure on a lamina held obliquely in a stream, and the position of the 

 centre of pressure. The distance of the centre of pressure from the 

 middle is ^ Z cos o / (4 + tt sin a), where a is the inclination of the 

 plane of the lamina to the direction of the stream. For a = ; this 

 divides the breadth in the ratio 11:5; consequently, a blade swinging 

 about an axis nearer the front edge than 5/11 the breadth will be in 

 stable equilibrium ; if further from the front edge, the stable positions 

 of equilibrium will be inclined at angles o, on either side, given by the 

 above formula ; whilst, if pivoted at the centre, the only position is 

 perpendicular to the stream. He finds that the greatest pressure transverse 

 to the stream is for an inclination nearly equal to 39". 



The problem of the vena contracta is a very old one, and is a case 

 of the discontinuous motion we have been noticing. Hanlon^ and 

 Maxwell'' have applied the principle of momentum to its consideration ; 

 the former deduced that the contraction must be '5, whilst the latter 

 showed that it must slightly exceed this. Rayleigh,^ by slightly varying 

 the circumstances, and supposing the fluid issuing from a nozzle of 

 sufBcient length projecting into the fluid, has deduced that, for this case, 

 the coefficient of contraction is almost exactly -5. Considei'ing also the 

 case of fluid issuing from a finite cylinder through a similar nozzle, he 

 has shown that the section of the nozzle is a harmonic mean between the 

 sections of the cylinder and the jet. 



It is by no means easy to illustrate rigorously, by experiment, all the 

 results deduced from the foregoing theory ; for, apart from the extreme 

 instability which is a general characteristic of jets and allied motions, 

 other disturbing influences arise from capillary action, in the case of jets 

 into another fluid or vacuum, which tends to break up a column into 

 drops, and from fluid friction, even in the case of two portions of the 

 same fluid moving past each other, where the influence of surface-tension 



* See under special problems. 



2 British Assoc, Glasgow; also, 'On the Resistance of Fluids,' PMl. Mm. (5) ii. 

 p. 430. 



5 ' The Vena Contracta,' Proc. Lond. Math. Soc, iii. p. 4. 



* Remarks on the preceding paper, ih., p. 6. 



* 'Notes on Hydrodynamics,' Phil. Mag. (5) ii. p. 441. 



