the Robinson Cup- Anemometer. 69 



to the application of Lord Rayleigh's formula, in addition to 

 those previously considered, viz. that the stream-lines cannot 

 at once assume the positions given them by an equilibrium- 

 theory. 



Taking the equilibrium- values in default of better, we obtain 

 the resultant force and couple per unit length of plank by 

 writing U 2 for V 2 and -ty for a in (1) and (3). The wind's 

 action, it should be noticed, no longer aids the motion through- 

 out a whole half revolution, but only through an angular 

 interval cos" 1 (v/V) on each side of Oy. As v/Y increases 

 this angle continually diminishes, vanishing of course in the 

 limit when v=Y. 



When b/l is not small the problem, even from the equi- 

 librium standpoint, becomes much more complicated. In the 

 positions A and B in the figure, the velocity of the wind 

 relative to the plank decreases as the distance from increases, 

 the reverse holding in the positions C and D. The natural 

 thing might seem to be to assume Lord Rayleigh's result as 

 holding for successive narrow parallel strips of the plank, and 

 then integrate * the elementary couples so found across the 

 width b. It is quite possible the result thus obtained might 

 be fairly accurate, but it is perhaps quite as likely it might 

 not. Until the problem actually presented has been satis- 

 factorily solved, certainty cannot be reached by any amount 

 of general reasoning. 



So long as b/l though not negligible is decidedly small, I 

 suspect the best one can do in our present state of knowledge 

 is to take for the resultant pressure and couple the values found 

 by regarding U as everywhere the same as at the central line. 



It may perhaps be as well to point out that as the relative 

 velocity, for a uniform v, is greatest when the motion is against 

 the wind, the wind's action necessarily tends to diminish an 

 initial velocity in a symmetrical body like a plank with equal 

 plane faces. 



§ 5. All the difficulties we have noticed in connexion with 

 the imaginary infinite planks present themselves in the case of 

 the Robinson cups, and most of them in an accentuated form. 

 The pressure experienced by a hemispherical cup moving 

 through a perfect liquid has not been determined mathemati- 

 cally, even for the simplest case when the wind's direction is 

 along the axis of the cup, i. e. is perpendicular to the plane base 

 of the hemisphere. The absolute velocity, answering to the 



* A process similar to this is adopted by Prof. G. H. L. Hagen in a 

 paper in the Abhandlungen of the Berlin Academy, the original of which 

 I have not seen. This paper is reprinted in Prof. Cleveland Abbes 

 < Mechanics of the Earth's Atmosphere' (see p. 20). 



