76 



Mr. C. Chree on the Theory of 



start is given by 



dv 

 dt 



= -« -6 1 V + c 2 Y 5 



(17) 



If the wind is high the term c 2 Y' £ must preponderate, if c 2 be 

 not zero, and v must be positive as the cups always move 

 convexity first : thus c 2 must be positive. 

 Consider next the equation 



a + a l v + b i Y + a 2 v 2 + 2b 2 vY-c 2 Y 2 = 0, . . (18) 



giving the velocity v in the steady state answering to a wind 

 of uniform velocity Y. If the wind be high, Y/v is very 



approximately deducible from 



c 2 (Y/vy-2b 2 (V/v)-a 2 =0. 



(19) 



Now for any given value of Y experiment shows that a 

 steady state is possible. Thus (19) must have a positive root. 

 Now Y/v is certainly not less than 2, and so if b 2 were nega- 

 tive it would be necessary for a 2 to be at least 4 times the 

 numerical sum of b 2 and c 2 to allow of (19) having a suitable 

 positive root. Professor Stokes*, however, in treating of 

 Dr. Robinson's results, decided that they fitted a formula 

 such as (19) best when in it a 2 /b 2 and a 2 /c 2 were supposed 

 small or even zero. The alternative that b 2 is positive is thus 

 much the most probable. If a 2 , b 2 , c 2 are, as we suppose, all 

 positive, there is but one change of sign on the left-hand side 

 of (19), a result in accordance with the observed fact that for 

 any given wind- velocity there is only one steady state. 



The term — a in (14) can hardly answer to anything but a 

 frictional force, presumably between metal surfaces, opposing 

 the motion, so there can be little doubt a is positive. 



As to the signs of a ± and 5 X there is more uncertainty. If 

 a x were a negative quantity, then, supposing a 2 small, the 

 right-hand side of (1 6) might become positive when v lay 

 within certain limits ; this would imply that if, during a 

 calm, the cups were given a suitably selected initial velocity, 

 this velocity would for a time go on accelerating. Such a 

 phenomenon is, to say the least of it, most improbable. The 

 alternative that a x is positive seems thus the most probable, 

 and should certainly be adopted if there is reason to believe 

 a 2 zero or very small. 



Either sign for A x seems to suit all the mathematical criteria 

 equally well, and there is nothing on the physical side to turn 

 the scale. It fortunately does not seem of any importance to 



* Phil. Trans, for 1878, pp. 820-1. The formula actually given by 

 Prof. Stokes includes a term answering to a in (18). 



