284 Prof. Morton and Mr. Vint on the Paths of 



the time integral of magnetic force, but proportional and 

 perpendicular to it ; this seems to be the explanation of the 

 difficulty noticed by Sir Joseph Larmor in the last complete 

 paragraph on page 705, Phil. Mag. Nov. 1914. The con- 

 tinuity of magnetic force normal to the reflector is due to the 

 alteration in the angle of reflexion, so that m' sin 0' = m sin 0. 

 The assumption that the electric intensity along the ray is 

 constant, and is not altered by reflexion, is essential to the 

 argument given above. 



XXVI. On the Paths of the Particles in some cases of Motion 

 of Friciionless Fluid in a Rotating Enclosure. By W. B. 

 Morton, M.A., and J as. Vint, M.A., Queen's University, 



Belfast *. 



[Plates J. & II.] 



IN a paper by one of us (Proc. Boy. Soc. vol. lxxxix. 

 p. 106, 1913), drawings were given of the paths of the 

 liquid particles in some cases of two-dimensional motion. 

 The present communication contains some notes supple- 

 mentary to the former treatment of the cases of rotating 

 cylinders containing liquid, followed by a discussion of some 

 three-dimensional paths which occur in the rotation of an 

 ellipsoidal shell containing liquid. 



Paths in rotating cylinders. — The stream-function yfr — 

 Qr n cos n6 gives the motion when the form of the containing 

 cylinder and of the relative stream-lines within it are 

 given by 



O n cos nO — ^cor 2 = const. 



The cases n = 2 and n = 3 were discussed in the former paper. 

 The case w = 4 can also be treated by use of elliptic functions. 

 It corresponds to the rotation of a cylinder having square 

 symmetry, and has been treated by St. Venant for the 

 elastic torsional interpretation of i|r : particulars are given 

 in Thomson & Tait, part ii. p. 244. For properly chosen 

 values of the constants we get a very close approximation to 

 a square section, the corners being slightly rounded off: 

 another case is that of a section bounded by hyperbolic arcs. 

 We have drawn a number of paths of particles in these cases, 

 but it does not seem worth while to reproduce these, their 

 general form being quite similar to those already given for 

 the triangular prism. The angular width of the loops on the 

 paths shrinks to zero at the centre. This will always be the 



* Communicated by the Authors. 



