314 PROCEEDINGS OF THE AMERICAN ACADEMY. 



Whence 



(U) 



1 ± kW^ - 1 



1 ± /t^4 y;3'/?4 



This equation (44) gives the relation of the wave lengths to the 

 parameter k, and is applicable only provided fJ has its optimum value 

 and T^ > r]zr]i. The condition that T have its optimum value is 

 conveniently expressed in the form of equation (36) above. Equations 

 r44) and (86) must both be used with the same sign in order to be 

 simultaneously correct. 



In case t^ < 7/3^74, equations (44) and (41) cannot be employed. In 

 this case Uopt and Vopt are both zero (see page 302), and a special 

 investigation is necessaiy. Tliis proves to be simple. Let us take 

 equation (16), make (/ = U^pt = 0, and we have 



(4y) Yiu ,) = — , 



Expressing this result in terms of Ymax max by dividing equation (45) 

 by equation (23), we have 



(«) (^)= M^l^M 



\ ymax max J (J/ 2 ,,,2 _^ y^ j^^^2 _^ J^2 / ^^.^ " ^ j 





Ki.-jn-v^{-a)T 



Equation (46) is to be employed in place of (41) whenever 



An interesting case arises when T^^/r/gT/i — 1. Equation (46) then 

 simplifies to 



^ Ymax max ' 1 i 1 _ Z' ^ ^^ ' ^ 



4vr \k) \ 



