62 Proceedings of the Roijal Irish Academy. 



I. A cylinder of water of indefinite length, originally at rest, is 

 set and maintained in motion by a constant spin O applied to the 

 bounding surface to determine the state of motion of the water at 

 any time t. 



Assume C = f2 + ^c^^e'^^'^JQ^a^r), since, for ^ oo , ^ = O ; 

 we have, then. 



We have now the following conditions : — 



(1) . For r = a, at all times to = fl. 



This condition will be satisfied by cancelling the term F{t), and 

 taking the am as the non-zero roots of Ji{aa) = 0. 



(2) . For t = 0,r between and t?, w = 0. This gives 



«r + 2S--/i(a„r) = 



between these limits. Multiply by rJi(a^r), and integrate between 

 and a ; then 



n r r'J,{a„,r)dr + 2 — f " rJ,\a„,r) = 0. 



Jo 0.fn J Q 



Now, we have 



fa ^2 



r« 1 



(a„,r) dr = a^J^ia^a) ; 



Jo 



n 



whence, finally, 



(or = Or + 20S^^"Vi(a^r) /a„.Jo(a^<?). 



As t becomes large, we may retain only the first term of the S ; also, 

 we have, q. p., 



Stt 



Aa 



