G. E. Curtis — Theory of the Wind Vane. 47 



both wings in opposite directions, but when d is equal to or 

 greater than e, the wind will strike only one wing as in the case 

 of a straight vane. Therefore, to obtain a measure of relative 

 sensitiveness, we must compare the gyratory force acting in 

 each of these two cases with the corresponding force acting on 

 'a straight vane. For 'simplicity, suppose the air between the 

 wings to be unaffected by the wind, as would be the case if the 

 space were entirely enclosed. 



1. When d < e. The gyratory force is proportional to 

 P[sin(£+#) — sin(e— 0)] = P sin d . 2 cos e, in which, for all ordin- 

 ary angles of the wings, 2 cos £ may be put equal to 1*9. To 

 compare this expression with P sin # (1 + w), the expression 

 above given for the corresponding gyratory force acting 

 upon a straight vane, we must know the approximate value 

 of n, which varies with the size and shape of the plate. Ex- 

 periments on normal planes of small size give values of n 

 ranging from 0*2 to 0*86 ; for surfaces as large as ordinary 

 wind vanes, the upper limit of n can hardly be as great as 

 0"5 and probably is less. With this value, the ratio of the 

 gyratory force acting on the straight vane to that acting on 

 the spread vane, when d <£, is 



1-5:1-9 



2. When d= or > e. The wind strikes the exposed surface at 

 an angle d + e and passes the sheltered surface at an angle d—e. 



The total gyratory force, therefore, will be 



Psin(0+£) + wPsin(0-£). 

 Comparing this expression with P sin #-f-wP sin d, the 

 corresponding gyratory force upon a straight vane, the first 

 term is seen to be larger, and the second term smaller, than the 

 respective terms of the latter formula. 



If - 5, the maximum value of n, be substituted in these 

 formulas, the difference between the two forces will be 



w = -| sin £ cos d— f (1 — cos e) sin 6, 

 which is the excess of the force upon the spread vane over that 

 upon the straight vane. For all values of £ less than 41 -°4 this 

 excess is positive and has its greatest* value when d=e; with 

 the increase of d, the excess diminishes and becomes at differ- 

 ent points depending on the value of e, thus 



for £=10 - ° the excess vanishes for 6= f j5°'S 

 £=20- " " 0=62°-l 



6 = 30- " " 6=5l°-2 



6 — 4.1-4 « « tf=41°-4 



* By an analytical investigation of the nature of u considered as a function of 

 the two independent variables, 8 and e, it appears that this function has no 

 critical value for ranges of e and 8 between 0° and 90° ; but the greatest positive 

 value of u (not a critical value) will occur when 8=e, the limiting value of 8 under 

 the present case. 



