the Robinson Cup- Anemometer. 79 



and of the same order of magnitude as that of the wind. If 

 V/v be practically constant for different values of V, then vfK 

 is so likewise. 



If, on the other hand, the frequency be high and V not too 

 big, a tends towards tt/2, while v becomes small. This 

 signifies that the time when the cups move fastest tends to 

 approach that when the wind has slowed down to its mean 

 value, but the variation in the cups' velocity is proportionally 

 much less than that in the wind's. In this case the action of 

 the Robinson cups is conspicuously to smooth down the varia- 

 tion actually occurring in the wind's velocity, without intro- 

 ducing appreciable error into the measurement of the total 

 space travelled by the wind. 



If the expression for the wind's velocity contained a series 

 of small harmonic terms of different frequencies, conclusions 

 similar to the above would equally apply to each term. 



General Solution when a 2 is zero. 

 § 12. When o 2 is zero the integral of (14) is known to be* 



t , = ,-/o>+^)* {0 + j- (c2 v2_j 1 V-« )/ ( "' +2W '"^}, . (27) 



where C is a constant determined by the initial conditions. 



When V is a known function of t, v can be evaluated to 

 any required degree of accuracy by quadratures, but the 

 operation will generally be tedious. 



As an example, suppose that after the steady state answer- 

 ing to Y = A has been attained, V diminishes according to 

 the law 



Y = A-Bt, 

 where B is a constant. 



Substituting for V in (27), and determining so that 

 when t = the value of v is that answering to a wind- velocity 

 A in the steady state, i. e. is given by 



v=(c 2 A 2 -b 1 A-a )/(a l + 2b 2 A), . . . (28) 

 we find 



c 2 A-^ c 2 a x c 2 Bt ( c 2 a? «A_ y-{(«.+«M>'-V"*} 

 2b 2 4V 2b 2 + V 46 2 2 + 2b 2 a °) a x +2b 2 A. 



* Forsyth's ' Differential Equations,' p. 18. 



