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



tion of P from the axis. Then the differential equation of 

 motion for a straight vane will be 



(V0 j. . I . . r dd\ 



where K^'f* ' r . 



When d is small, sin 6 may be replaced by d, and simplify- 

 ing the constants by making 'Kv"' = (o\ and ~Krv=2k the equa- 

 tion becomes 



This is the equation of a circular pendulum in a resisting 

 medium where the resistance varies as the velocity. The gen- 

 eral integral of this equation takes three forms according as 

 h' — af is positive, negative, or vanishes; a numerical evalua- 

 tion shows that it is always negative, for which case, the inte- 

 gral becomes 



— Jet ( . Jc . . 

 0=ae i cos ht + - sin lit 



where a is the greatest value of d and h" = a) i — k-. 



Por a spread vane, where the half angle of the wings is greater 

 than the deviation of the wind, the wind acts on both wings at 

 once in opposite directions, and the resultant pressure will be 



• , ns r d0 T • / zn r d0 "I 



in which -j- is to be taken with opposite signs in the two terms, 



and the expression reduces to 



c ■ n *r dd 



2 sin o cos e + 2 — tt- 

 v at 



P Using the same notation as in the previous case, and making 

 2 cos £=1*9, which is approximately true for all ordinary cases,' 

 we have the equation • 



^ +4&^- +l-9c» a 0=O. Making h/=V9w*-4Jc\ 



df dt ° ' 



— iJct { 

 )=ae j cos ht + 



2Jc . ■ _ } 

 T *nhfi\ 



d6 h* + W -2Jct . . 



-=- = — a . - J —^ . e . sin h t. 



dt h r 



dd 7t 27T 



-v-=0, when t=0, t— -rt= ~y-, etc. 

 dt h, h, 



