Lord Rayleigh on the Resistance of Fluids, 435 



(3) for sin 2 a will materially modify the angle at which the 

 transverse force is greatest. The quantity to be made a maxi- 

 mum is 



sin a cos « _ 



4 + 7Tsina' 



and the value of <x for which the maximum is attained is « == 39° 

 nearly, being considerably less than according to the old theory, 

 on account of the increased value of the normal pressure at 

 high obliquities. 



The pressure, whose mean amount is given in (3), is far 

 from symmetrically distributed over the breadth of the blade, 

 as might be anticipated from the fact that the region of maxi- 

 mum pressure, where the stream divides, is evidently nearer 

 to the anterior or up-stream edge. If the breadth of the blade 

 be called /, the distance (a) of the centre of pressure, reckoned 

 from the middle, is 



3 cos a / 



a 



4c 4 + 7r sin a 



(±) 



If the blade be pivoted so as to be free to turn about an 

 axis parallel to its edges, (4) gives the position of the axis cor- 

 responding to any angle of inclination a. If a = 90°, <£ = 0, 

 as is evident from symmetry. As a diminishes, the corre- 

 sponding value of x increases and reaches a maximum, viz. 



3 



— I, when a = 0. The axis then divides the breadth of the 



16 



blade in the ratio 11 : 5. 



These results may be stated in another form as follows. If 

 the axis of suspension divide the width in a more extreme 

 ratio than 11:5, there is but one position of stable equilibrium, 

 that namely in which the blade is parallel to the stream with 

 the narrower portion directed upwards. If the axis be situ- 

 ated exactly at the point which divides the width in the ratio 

 11:5, this position becomes neutral, in the sense that for small 

 displacements the force of restitution is of the second order, 

 but the equilibrium is really stable. When the axis is still 

 nearer the centre of figure, the position parallel to the stream 

 becomes unstable, and is replaced by two inclined positions 

 given by (4), making with the stream equal angles, which 

 increase from zero to a right angle as the axis moves in 

 towards the centre. With the centre line itself for axis, the 

 lamina can only remain at rest when transverse to the stream, 

 though of course with either face turned upwards. 



The fact, rather paradoxical to the uninitiated, that a blade 

 free to turn about its centre line sets itself trail sverselv, may be 



2 F 2 



