G. E. Curtis — Theory of the Wind Vane. 49 



by Parrot for his spread vane, and has been the justified claim 

 of his followers. Given two vanes of the same moment of 

 inertia, for slowly shifting winds, the oscillations of the spread 

 vane will be smaller than those of a straight vane, for, being 

 more sensitive, it responds more quickly to any gradual change 

 in wind direction, and, therefore, has a less amplitude of oscil- 

 lation, and will sooner come to rest. When the wind suddenly 

 changes direction through a considerable angle, the straight 

 and spread vanes start out with the same angle of deviation 

 from the new direction. The following analysis shows that, in 



this case also, the spread vane comes 

 sooner to rest. If v be the velocity 

 of the wind, r a constant depending 

 on the length of the vane, d the angle 

 between the direction of the wind and 

 the axis of a straight vane, the result- 

 ant relative velocity, K, of the wind 

 and the vane will be the third side of 

 a triangle of which two sides are the 

 velocities of the wind and vane, and 

 90° — #, their included angle. 



ttti ™ . . « dO . a / dd\ 2 dd _ . 



Whence K =v +2v r— sin o+ 1 r~) , — being negative or 



positive, according as the motion of the vane is with or against 

 the wind. This resultant makes an angle <p with the vane such 

 that 



. a dd 



K sin cp=v sin o +r—^- i 



R cos g>=v cos 0; 



The effective force tending to turn the vane, which is pro- 

 portional to v a sin 6 when the vane is at rest, becomes R 2 sin (p 

 when the vane is in motion. Considering the motion when 6 and 



-5- are so small that we may neglect the square of -=- and the 



. o d0 



product, sin u -j-, 



3L r dd 



R 2 becomes v 2 , and sm cp= sin 6+ - . -—, 

 ' 9 r v dt' 



Whence R a sin <p=Wsin 6 + - . -j-). 



Let I be the moment of inertia of the vane around its axis, 



p the air density, A the surface of the vane,/ (P) the pressure 



of a wind of unit velocity on a unit surface under a standard 



density of the air p , and r the distance of the point of applica- 



Air. Jour. Scl— Third Series, Vol. XXXIV, No. 199.— July, 1887. 

 4 



