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



For a smaller value of ?i, the excess would vanish at larger 

 values of 6. 



Consequently, for all moderate values of e and 6 when d <£, 

 as well as when 6 <e, the gyratory force acting on the spread 

 vane is greater than that on the straight vane. 



If a perfectly sensitive vane be defined as one that instantly 

 responds to the slightest change in the direction of the wind, 

 that vane upon which the winds exert the greatest gyratory 

 force will be the most sensitive, supposing the vanes all have 

 the same friction. The increased sensitiveness accruing to the 

 spread vane from its greater gyratory force is diminished and, 

 for large values of e, may be quite overcome by its greater lat- 

 eral friction, due to that component of the wind pressure which 

 passes through the axis transversely. This component is pro- 

 portional to sin 2 e, nearly, and so its effect, which is small and 

 negligible for small values of e, increases rapidly and becomes 

 an important factor for values of s greater than 20° or 25°. 

 This lateral friction, therefore, constitutes a condition requiring 

 the angle between the wings to be small. A further indication 

 as to the angle at which the wings should be set for maximun 

 sensitiveness is found by an analysis of the expressions given 

 above for the gyratory force. 



1. When e= or > 6. Place s=d+a; then 



2 sin 6 cos e=2 sin d cos (d + a). 

 Noting that (d-\-a) must always lie in the first quadrant, this 

 expression has its greatest value when a=0, i. e. when the half 

 angle of the wings is not larger than the angle of the wind's 

 deviation. 



2. When e < d. Examining the expression for gyratory 

 force, sin (d + e)+n sin (#—£), to find a critical value, it appears 



1 — n 



that the function is a maximum when tan £=r— — cot #. 



1-t-n 



Making n==% as before, tan £=-§■ cot 6. For all values of d up 

 to 30°, this makes the corresponding value of e to be greater 

 than d, which for the present case is an impossible value ; but 

 for this range of d, from 6° to 30°, the greatest value of the 

 function (not a critical value) will occur when e=6. For val- 

 ues of d above 30°, the value of e giving a maximum dimin- 

 ishes downward from 30°, and for 0=90 becomes 0. Conse- 

 quently the best values for £ for deviation of the wind below 

 30° will hold good also for deviations greater than 30°. The 

 conclusion, therefore, is that, for maximum sensitiveness, the 

 half angle of the wings should be the average angle through 

 which the wind makes sudden deviations, this average being 

 taken from to 30°. 



The question of the relative stability of a spread and straight 

 vane remains to be discussed. That the continual oscillations 

 of the wind vane are diminished, was the advantage claimed 



