the Robinson Cup- Anemometer. 89 



When n is large the expression inside the last square 

 bracket cannot exceed the asymptotic value 



f , v' - v m + y"\ v'" - v") + y'"y"{y" - v') \ 



C(i - y f ) + cy (i -y") + C'V^'Ci-y") 



fr.JU.tf1 



1 ~yy y 



and thus is negligible compared to the first two expressions, 

 which contain b as a factor. Omitting it, we have a com- 

 paratively simple formula for the mean velocity 



feis(T 1 + T 2 + T 3 ). 



This formula, though probably the shortest for calculating the 

 numerical value of v in a given case, is less convenient for the 

 purpose of proving v in excess of the equilibrium value than 

 is the following formula, deduced from it by algebraical 

 manipulation : — 



g(Tj + T 2 + T 3 ) - («T, + f-"T 2 + v"%) 



= iJJX"' [C ' C " (1 ~ y,) (1 - y " ] (V'-V'0(e'-«") 



4 C'V/"(l-y // )(l-y f ")(V"-Y /// ) (y"-v<") 

 + C'C^l -y'") (l-^OT^-V 7 ) (« /// -« / )] 



1—yyy 



+ C'"(v'" -€'')]• • ( 65 ) 



Each of the three terms inside the first square bracket is, 

 as seen in a similar case before, necessarily positive ;' but one 

 at least of the three terms inside the last square bracket is 

 negative. 



The subsequent proof depends on the exact circumstances 

 of the case. Suppose for instance 



V>V">V", 

 then put 

 0\v f -v") + 0"(v f, -v fff ) + C'"(«'"--t; T ) 



= 2C / C%(V / - V") ( 0" - *?"') - 2C / C /// 6 2 (V '" - V) (s" 7 - v>) . 



The first term of the two on the right is positive according 

 to the hypothesis made. Now combine the second term with 

 that one* of the three terms in the first square bracket in (65) 



