Purser — Applications of BesseTs Functions to Fltytiica. 53 



Form of Issuing Jet, where steady motion has been attained. 



Consider a distance ^ below orifice conditioned as above ; and 

 let li^ denote the vertical velocity which may now be assumed 

 uniform over cross-section, the horizontal velocity also being 



now negligible. We have, then, ^ (w'^ - U^) = g {h ^ t)- Also, 



evidently UR'"^ = uBr = ?<'p-, where p is radius of cross- section 

 corresponding to ^. The equation to the curve of the jet is therefore 



or neglect 



2 {gh -f- 



Stream Lines and Radial Velocity. 



The vertical velocity or ^ at any point is given by 



d(ii 2uR . , . 2 _ 



— = —j— 2„ cos nl . KQ{nr) Yi{nR) sm nz + j + U 



(IZ c I 



for points in internal cylinder, and by 



d^ ^ 2uR YJinr)Ki{nR) sin nz + u[ \ - ^j 



for external cylinder. If we write 2 = ^ - ^, the former of these 

 expressions may be written 



and the latter 



provided R be sufficiently small with respect to / to allow us to 

 substitute integrals for sums. 



Similarly for internal cylinder 



