the Robinson Cap- Anemometer. 73 



known, the outcome of the mathematical work — which is of 

 considerable merit — is of somewhat doubtful practical utility. 

 At the same time the paper seems to show the line a complete 

 mathematical solution might take if fuller experimental know- 

 ledge existed. 



Deduction of an Equation of Motion. 



§ 7. I now propose to look at the matter a little differently. 

 The couples acting on the instrument at any instant are 

 primarily three : — A couple such as pIPR 2 // (yjr) in (b), where 

 /(>/r) may be in reality a function of R, R//, and v/Y. A 

 couple —(/3 + 7IP) arising from the friction opposing the 

 motion, where 7 is probably a function of R, R/7, and v/Y, 

 but not of the mass turning ; y6 is proportional to the mass 

 turning. A couple arising from the viscous action of the wind 

 which may be taken as opposing the motion. Provisionally 

 it may be supposed to be of the form 



-KV-K'y, 



where K and K' are constants. 



During a revolution of the cups v is usually nearly con- 

 stant — more nearly so than Y, most probably, as a rule. 

 Thus we may expect to approximate closely to the true motion 

 by treating the driving force as having at any instant the 

 mean value it would possess throughout a complete revolution 

 during which V and v retained unchanged their instantaneous 

 values*. The equation of motion so derived would be 



g= l^= ±^"[ P WWPfW)-l(l3+vW)-l(KV+K>v)]de. (12) 



If f(yjr) and 7 be assumed to contain only integral powers of 

 v/Y we should, assuming expansion possible, obtain an equa- 

 tion of the type 



^^-a -a 1 v-b 1 Y + Y 2 ^c 2 -2b 2 ^ -a 2 (^j "^(j) ~-\> (13 ) 



where a w a x , &c, are independent of v or V. 



Unless we know the forms of f{#) and 7 we cannot tell 

 whether the series inside the square bracket consists of a finite 

 or an infinite number of terms ; and if it be an infinite series 

 we have no data relative to its convergency. 



If the series were infinite it could hardly be convergent in 



* In the ordinary instrument with four equal cups, the entire cycle of 

 changes with respect to the wind's incidence is gone through every 

 quarter revolution, thus the hypothesis involves no greater assumption 

 than that changes in V or v during a quarter revolution are small. 



