74 Mr. C. Chree on the Theory of 



the form shown unless v were less than V. Now conditions 

 certainly arise when this is not the case. If, for instance, 

 during a calm the instrument were put in rotation by the 

 hand, and left to itself, there would initially be an infinite 

 value for v/Y, and somewhat similar phenomena must occur 

 during a sudden lull after a violent gust of wind. 



It is of course possible that one equation of motion holds 

 for values of v/Y less than unity, and a totally different equa- 

 tion for greater values. It is difficult, however, to imagine a 

 cause for such discontinuity. The wind, as we have seen, 

 always opposes the cup's motion during the greater part of 

 each revolution. As v/Y approaches unity the corresponding 

 physical phenomenon is the contraction of the angular interval 

 throughout which the wind aids the motion. The interval 

 simply vanishes when v/Y equals unity and remains zero for 

 all higher values of v/Y. There is no suggestion of discon- 

 tinuity in the physical conditions. 



The only simple form of equation suggested by the pre- 

 ceding investigations which is applicable without inconsistency, 

 whether v/Y be small or big, is 



dv 



dt 





a 1 v-b 1 Y-a 2 v 2 -2b 2 vY + c 2 Y 2 . . (14) 



It is assumed of course that neither v nor V can be negative. 

 The term — a answers to friction, the full value of which is 

 called into play only when the cups move or are on the verge 

 of motion. The critical wind-velocity V required to call up 

 the full amount of this friction, so as just to prevent motion, 



dv 

 is obtained by supposing v and — in (14) to vanish. The 



resulting equation is 



c 2 Vo 2 -&iVo-ao = (15) 



For values of V less than Y only so much friction is called 



dv 

 into play as suffices to prevent motion. The possibility of -j- 



being negative, when v is zero, is thus precluded. 



§ 8. The reasoning by which we have been led to (14) 

 cannot claim to be conclusive. The formula, though to a 

 certain extent suggested by theory, is admittedly in part 

 empirical, and must be accepted only provisionally. The 

 following are perhaps the strongest points in its favour ; — 



1. It is, as explained above, strictly in accordance with the 

 observed fact that the cups do not move until the wind has 

 attained a certain velocity. 



