46 G. E. Curtis — Theory of the Wind Vane, 



sultant normal pressure upon the surface proportional to sin a. 

 This was based upon the theory that the amount of fluid im- 

 pinging on a unit of surface is proportional to sin a, and the 

 normal component of pressure exerted also proportional to sin 

 a: or again, the velocity of the fluid resolved perpendicularly 

 to the surface being v sin a, the resultant force exerted against 

 the surface must be proportional to v 1 sin'a. But all this 

 reasoning is now known to be worthless. Experiment shows 

 that the resistance is much more nearly proportional to the sin a 

 than to its square. Colonel Duchemin* was one of the first to 

 give a more accurate formula based upon the results of experi- 

 ments. He found that the pressure on a plane moving 

 obliquely in a fluid may be represented by the following em- 

 pirical formula. 



2 sin a 

 1 + sin a 



in which P is the resistance to a plane moving normally. Later 

 experiments, made for the London Aeronautical Society in 1872, 

 indicate that for large values of a the observations would be 

 better satisfied by making the denominator 1+sinet instead of 

 l + sin 2 a. This change has also received a theoretical confirma- 

 tion. By the analytical method of Kirchhoff and Helmholtz, 

 Lord Rayleigh has derived a formula for the case of a blade- 

 shaped surface which makes the pressure per unit of area 



7rsit pv 2 . For small values of a neither of these formulas 



4 + 7T sin a 



differ much from sin a, and for the purposes of this paper it will 



be assumed that this latter relation holds good : therefore the 



normal pressure on the surface of the vane tending to produce 



rotation will be proportional to the sine of the angle between 



the surface and the wind direction. 



Let (1+njP be the total effective pressure on a straight vane 

 when at right angles to the wind, where n takes account of the 

 diminution of pressure on the sheltered side. The value of n 

 varies with the angle of obliquity in some ratio not yet fully 

 known, but, for its first approximation may be assumed, like 

 the pressure on the exposed surface, to be proportional to the 

 sine. On these assumptions the gyratory force tending to 

 restore the equilibrium of the vane will be, for an angle d be- 

 tween the vane and the wind direction, proportional to (l + n)P 

 sin d. 



For a spread vane, let e be half the angle between the two 

 win^s and d the. angle between the medial line of the vane and 

 the wind direction. If d be less than e, the wind will act on 



* Duchemin : Recherches experiinentales sur les lois de la resistance des fluides, 

 Paris, 1842. 



