286 Paths of Particles in Motion of Frictionless Fluid. 



This magnitude is < 1.. The motion of a particle in space 

 is thus obtained by com lining simple harmonic motion round 

 an ellipse with a rotation of the ellipse itself round the axis 

 (hnn). 



To examine the simplest re-entrant forms of these space- 

 paths we take &>, Q in the ratio of small numbers. When 

 the value of this ratio is assigned a certain limitation is 

 imposed on the form of the ellipsoid. The axis of rotation 

 (I in n) must lie on the cone 



where A = 2bc/{b 2 + c 2 ), etc. 



Therefore (o/£l must lie between the greatest and least of the 

 magnitudes A, B, C. This is equivalent to saying that the 

 greatest and least of the axis-ratios of the ellipsoid (taken as 

 less than unity) must lie on opposite sides of the magnitude 

 (1— y'l — k 2 )jk where k = a>/Q,. The values of this are, 



for h = | -382 



•268 

 •172 

 •127 



These cases have been worked out numerically taking the 

 ellipsoid (5, 2, 1) for the first two and (10, 3, 1) for the 

 second two. In each case the cone which contains the 

 possible axes of rotation surrounds the mean axis of the 

 ellipsoid. The particular generator selected was such that 

 it made a considerable angle with the normal in the plane of 

 the ellipses, so that the path in space should have solid relief. 



Models of the curves were made in lead wire and stereo- 

 scopic photographs were taken of these. For & = §, J, J the 

 paths of particles lying on the central section only were con- 

 structed. These are shown on figs. 9, 10, 11 (PI. II.). For 

 /b~i the paths were calculated also for a series of sections at 

 different distances from the central plane, showing how the 

 path is continuously altered until it becomes a circle round 

 the axis of rotation for the particle on the surface situated 

 at the point where the tangent plane is parallel to the planes 

 of the paths relative to the ellipsoid. The successive forms 

 are given in figs. 12 to 16. 



The following is the detailed description of the cases of 

 motion for which the curves have been constructed. 



Fig. 9. Ellipsoid (5 2 1) rotating about the axis whose 

 cosines, referred to the axes of figure, are ('5£, *52, '62). 



