where 



78 Mr. C. Chree on the Theory of 



cease to be comparable unless the state of the lubricant 

 remains practically constant. 



Wind Velocity with small Harmonic Term. 



§ 11. Suppose the wind's velocity to vary harmonically 

 with the time according to the law 



V=Y(1+\cosj<), (21) 



where X is small. The wind's velocity runs through a com- 

 plete cycle in the time 27r/p, with maximum and minimum 

 values V(l + X) and mean value V. 



Assuming \ 2 negligible, we can satisfy (14) by supposing 



v=v{l-\-v cos (pt—cr)}, .... (22) 



a + a 1 v + b i Y + a 2 v 2 + 2b 2 vY--c 2 Y 2 =0, . . (23) 



o-=ttm- 1 p/{a 1 + 2a 2 v + 2b 2 V}, .... (24) 



r _.T (2^-2^-6,) m m . (25) * 



® ^p 2 + (2b 2 V + 2a 2 v+a i y 



From (22) we see the velocity of the cups has a mean 

 value v, and has a harmonic variation of the same frequency 

 pj2ir as the wind but different in phase. From (23) we see 

 that v is the velocity the cups would have in the steady state 

 answering to a wind of uniform velocity V. Thus the 

 deduction of the space traversed by the wind, in a time large 

 compared to 27r/p, from the value of j v dt in the ordinary 

 way is practically as satisfactory in this case as in that of a 

 steady wind. 



As to the details of the cups' motion, we see from (24) and 

 (25) that the phase and amplitude of the harmonic term in v 

 depend both on the frequency of the harmonic change in the 

 wind's velocity and the mean value of that velocity. 



If the frequency be small and the mean velocity great, 

 then approximately 



cr=0 L . 



r=x V ^g .[ .... (26) 



thus the harmonic motion of the cups is in the same phase 

 * If (14) be written ^ +/0, V)=0, then (25) may be written 



\ v dX p W ' 



