Particles in Motion of Frictionless Fluid. 285 



case when n>2, because then the relative paths approach the 

 circular form when r is small. 



For the elliptic cylinder, on the other hand, all the paths 

 are geometrically similar, and the angular width o£ the loops 

 remains the same right in to the origin. A point not noticed 

 in the former paper is that these paths are trochoids gene- 

 rated by the rolling of a circle of radius (a — b) 2 j2(a + b) 

 outside a circle of radius 2ab/(a + b), the tracing point being 

 at distance i(a— b) from the centre of the former circle. 

 This follows easily from the method of generation of the 

 curves. 



Paths in cylinders rotating about eccentric axes. — It is 

 obvious that the stream-lines relative to the cylinder are the 

 same in this more general case as when the rotation takes 

 place round the axis of symmetry. Further, the displace- 

 ment of the particle along the relative stream-line for a 

 given angle of rotation remains the same. Thus it is easy 

 to calculate the paths in space for rotation round any point, 

 once the case of rotation round the axis of symmetry has been 

 treated. The manner in which the looped path is modified 

 depends on the progressive change in the arrangements of 

 the normals which can be drawn from the centre of rotation 

 to the relative stream-line. We have worked out the case 

 of the elliptic cylinder and the triangular prism. Some 

 diagrams relating to the former are shown on figs. 1-8 

 (PI. I.). The dotted ellipse shows the position of the 

 cylinder relative to the axis of rotation which goes through 

 the centre of the looped path. It will be seen that for points 

 on the minor axis alternate loops shrink and disappear, while 

 for points on the major axis pairs of adjacent loops coalesce 

 as the centre of rotation moves away from the centre of 

 figure. 



Paths in a rotating ellipsoidal shell. — If the ellipsoid 

 (abc) rotates with angular velocity XI round the axis 

 (I m n) through its centre, then, as was shown by Lord 

 Kelvin, the particles of the contained liquid move, relatively 

 to the ellipsoid, round ellipses lying in the planes 



Ix my nz 



+ 79/ 9 . — *t+ tt-s — T9\ = const. 



a\b 2 + c>) ' b\c 2 + a 2 ) ' c 2 {a 2 -tb 2 ) 



Particles on the boundary move round the intersections of 

 their planes with the ellipsoid, other particles move round 

 similar ellipses. This elliptical motion is retrograde and 

 simple harmonic, having period 27r/a>, where 



a) 2 ; _ / 2bcl \ 2 f leam \ 2 / 2abn\ 2 



a*-\b*+<?) + W«y + V^T^v* 



