54 Proceedings of the Royal Irish Academy. 



for external cylinder 



= - 2 cos nz cos nl lAnrjAinJi + —r ; — > 



dr I ^ ^ 21 21 r 



giving in internal cylinder the stream line equation 



2 cos 7il - Fi hiR) rKx {nr) sin nz + %. rh + 27- = C, 

 i fi 21 2 



and in external cylinder 



^uR 1 r"^ U U 



2 cos - r (wr) sin W2 . {nR) ^ U- - - r-z + - R'h = C. 

 fl 2 2C 21 



R 



If y be small, the trigonometric terms may be replaced for 

 internal cylinder by 



and for external by 



If, now, we refer back to the expressions previously found for 

 the attraction-components of a circular disk over which matter is 

 uniformly distributed, we shall find that these are proportional both 

 for internal and external cylinders to the corresponding velocity 

 components in the fluid-motion problem. In particular, we see that 

 the radial velocity-comnonent at orifice 



which is infinite at edge, as it should be from general theory of fluid 

 irrotational motion, and vanishes for r = 0, 



Application to the Theory of Elasticitt. 

 K. — Torsion of Right Cylinder. 



Assume 



u = '^„^A^Jfmr) '-B{mz) ; 

 V = ~ ^„A,„Ji{t}ir)-^S{mz) ; 



2V = 0. 



