84 Mr. C. Chree on the Theory of 



wind -velocity are rapid. Supposing 



a 2 (v[ + v , 2 )T 1 and a 2 (v'^ + v^)T 2 

 small, but n so large that 



na 2 [v[ 4- v l 2 ) T : and na 2 [v" + v£)T 2 



are large, we find for the mean velocity in the final steady 

 state, 



« s («*.+wV( T .+ T ») 



_ «;T 1 + t>,"T 2 foj + tflT. + fof + tflT, 

 Ti+T, + ~ 2(T!+T 2 ) 



r( ^^sivi-ViKK-k) \* "I 



x Lt 1+ (( B ;+»i)T I +w+<)T 1 )«; _1 J- (52) 



To deduce the true physical significance of (52) we must 

 determine the sign of 



Now by (36) we have 



vl~v[ = (a 1 + 26 2 V')/a 2 , 



v»-v»= (a 1 + 2b 2 V»)/a 2 ; 

 whence 



^-,» = W-<+^(V'-V"). . . (53) 



Now the velocity of the cups in the steady state answering 

 to a uniform wind is unquestionably greater the stronger the 

 wind; thus V'— V" and v[— v" must have the same sign. 

 Consequently, assuming b 2 /a 2 positive, ^[ — v^^v^ — v^) has 

 the same sign as {v[ — v") 2 , or is necessarily positive. 



Thus we see that (52) implies that, under the conditions 

 supposed, whatever be the values of V'/V" and Tj/T 2 , the true 

 mean velocity v of the cups is in excess of 



WT 1+ <T 2 )/(T, + T 2 ), 



the value given by the equilibrium theory. The anemometer 

 trace, interpreted in the usual way, thus exaggerates the 

 mean velocity of the variable wind. This exaggeration in- 

 creases with v[ — v[ f , and so with V — V". The consequent 

 error would thus be most serious during a gale in which 

 violent gusto alternated with nearly calm intervals *. 



* See an interesting paper by Mr. R. H. Curtis, Quart. Journ. Roy. Met. 

 Soc. vol. xx. pp. 262 et seg. f especially pp. 266-7. 



