92 Mr Greenhill, On Conjugate Functions, &c. [Mar. 21, 



then since, expanded in a Fourier series, 



u 



T y „ „ COS mr-j-r 



K — E ir ^ A 



Sn U = -T-S-V5- TTTFF5 Z1I ' 



therefore 



k*K Jc 2 K 2 **" , K' ' 



COSh ?17T -|T- 



o cos mr %. cosh n-n- 4$. 

 J?* A A 



mp Zn ; ir 



cosh mr -~- 

 A 



„ sin mr -f>sinh mr 4* 

 ir 2 ^ A A 



y ~ ¥K iXn ; T' ■ 



cosh mr -~ 

 A 



and therefore if 



£ • , a • i V — ft 



sin mr ~ sinh mr T rSinh mr 





tfK 2 , K' . . a-/3 



cosh mr -^ smh mr — ~— 

 A A 



then i|r = when 77 = /S, 



ty = JJy when 77 = a, 



and ^ + Tv 2= ° ] 



and therefore ^ is the current function for the motion of the 

 liquid between the cylinder, 77 = a and 77 = /?, when the cylinder 

 77 = a is moving parallel to the axis of x with velocity U and the 

 cylinder 77 = /3 is fixed. 



If ^ denote the velocity function of this motion, being the 

 conjugate function to yjr, 



? . , a , 77-/9 

 , rr cos mr i>smh W7r -^cosh ?J7r t , 

 ,_ir U ^ A A A 



cosh mr -f= smh ?27r — ^r— 

 A A 



In a similar manner the current and velocity functions due 

 to any motion of translation or rotation of any of the cylinders, 

 £ = constant, or 77 = constant, may be written down. 



(2) On a Mathematical Law of Interest in Political Economy. 

 By Dr Akin-Karoly. 



