February 3, 1910] 



NATURE 



409 



known {e.g. Lanchester, "Aerodynamics," § So). If the 

 bounding surface meets the plane boundary OP at O, and O 

 is the origin, then dxv/dz = o when 2 = 0, whence we easily 

 have a = fev'3 in the above expression. 



Now substitute s=\'z', and the solution is obtained of a 

 continuous motion round the straight edge past O, with a 

 single vortex in the dead water in the wake of the edge 

 (Fig 2). 



Making the substitution, we now write 



. ..=A(s/--'+2//,iog V;'-<v;3^' |), 



V ^'= -(\/3-')*/ 



where A 



dwjdz' = - 



-(. 

 a constant, and we find that when s' = o. 



A ■J lb. 



A 



p 



O 



The velocity is thus finite where the stream leaves the 

 plate. 



With z' = z- we get a streaming motion round a rectan- 

 gular corner with a single vortex in the dead water (Fig. 3), 

 and similarly with s' = s", where K1K2, we get a stream- 

 ing motion past a projecting corner with re-entrant angle 

 n.i8o° (Fig. 4). But here comes the difficulty, if it is a 

 difficulty. 



Except in the above case of n = 2, the velocity vanishes 

 at the origin, and, further, the stream line bounding the 

 dead water makes equal angles with the two parts of 

 the fixed boundary ; thus, for the right angle of Fig. 3, 

 the boundary of tl?s dead water starts from the origin at 

 an angle of 135° with the two walls, and the dead water 

 projects forward into the stream. 



But is it not the fact that when a stream flows through 

 the arches of a bridge, the dead water does project into the 

 current, the circulating fluid pushing the stream into the 



/0'-- 







■■■Q 



centre and narrowing it? I believe I have seen something 

 of this very kind. 



As regards the velocity being zero, the same would occui 

 in the hydrodynamical problem representing the motion ol 

 two streams meeting at an angle, the velocity vanishinj: 

 at the projecting angle of the boundary. 



If, finally, we apply Schwarz and Christoffel's trans- 

 formation to our original figure, we can obtain various 

 solutions representing continuous motions past projectinj: 

 obstacles, maintained by a fixed vorte.x in the dead watci 

 behind them. For example, taking 



"^=C- 



'(s-0*(: + O*' 

 or (say), 



z'=.^{z^--c% 

 NO. 2I0I, VOL. 82] 



we get the solution for a broad stream with a pier project- 

 ing at right angles to the straight bank, or a current 

 impinging perpendicularly on a lamina, with a couple of 

 vortices situated in the dead water behind it. Moreover, if 

 c<2a, the whole of the back of the plate will be in the 

 dead water (Fig. 5), while if c>2a the current will flow 

 round and on to the plate, leaving dead water only near 

 the edges (Fig. 6). 



The whole point which I wish to emphasise is that hydro- 

 dynamical solutions can be obtained of cases of eddy 

 formation in the wake of a projecting obstacle by taking 

 Fig. I and the corresponding formula, and transforming 

 by the usual methods of conformal representation, trans- 

 fonning the point O of Fig. i into the projecting or 

 re-entrant angle. No other point can be so transformed 

 without making the velocity infinite, except P. We should 

 then have the vortices in front of the obstacle, and this 

 would certainly give a solution of the hydrodynamical 

 equations, but it is difficult to see how vortices would get 

 to the right points, and uncertain whether they would 

 be stable there. 



I have seen nothing like these solutions, yet it is hard_ to 

 imagine that anything so simple can have escaped attention 

 in a well-worn subject like hydrodynamics, especially as the 



{ o 



o 



o 



o 



motions bear a strong resemblance to certain observed 

 phenomena. If it should transpire that these problems have 

 b-en solved before, it seems desirable that attention should 

 be directed to them in view of the importance which such 

 problems have assumed in connection with aerial and otln-r 

 navigation. G. H. Bkv.in. 



THE NEW COMET (1910a). 

 I N those places where there has been a clear horizon at 

 sunset during the past week, the new comet has pro- 

 vided a striking spectacle for thousands of observers. The 

 observations made at the established observatories will 

 have to be reduced and discussed, and some time will 

 elapse before they are generally available, so at present we 

 have only the meagre details of telegraphic summaries. 



From these we learn that excellent photographs have 

 been obtained at Oxford, Cambridge, Dublin, Stonyhurst, 

 and other observatories, including the Harvard, Yerkes, 

 and Lick institutions. Numerous observers have recorded 

 changes in the appearance of the comet, and it will be 

 interesting to see if these are shown on the photographs. 



The elements and ephemeris issued from Kiel are 

 evidently considerably in error ; according to Prof. Turner, 

 the error was 3° in declination on January 26, and was 

 increasing 40' dailv. On that dav the comet's position was 

 determined at 5.35 p.m. by Dr. Rambaut, at the^Radchffe 

 Observatory, as R.A. = 2ih. 20m. 40s., dec. = 2 17 S. ; 

 according to the ephemeris, it should have been approxi- 

 mately 2ih. 26.9m., 0° 52.5' N. According to Mr. 

 Crommelin, speaking at the British Astronomical Associa- 



