258 ON A DYNAMICAL TOP. 



path of the instantaneous axis in space, the hoop that of the same axis in the 

 body, and the axis of the stick the invariable axis. The point of contact repre- 

 sents the pole of the instantaneous axis itself, travelling many times round the 

 stick before it gets once round the hoop. It is easy to see that the direction in 

 which the instantaneous axis travels round the hoop, is in this case the same as 

 that in which the hoop moves round the stick, so that if the top be spinning in 

 the direction L, M, N, the colours will appear in the same order. 



By screwing the bob B up the axle, the difference of the axes of inertia 

 may be diminished, and the time of a complete revolution of the invariable 

 axis in the body increased. By observing the number of revolutions of the top 

 in a complete cycle of colours of the invariable axis, we may determine the 

 ratio of the moments of inertia. 



By screwing the bob up farther, we may make the axle the principal axis of 

 least moment of inertia. 



The motion of the instantaneous axis will then be that of the point of 

 contact of the stick with the outside of the hoop rolling on it. The order of 

 colours will be N, M, L, if the top be spinning in the direction L, M, N t and 

 the more the bob is screwed up, the more rapidly will the colours change, till 

 it ceases to be possible to make the observations correctly. 



In calculating the dimensions of the parts of the instrument, it is necessary 

 to provide for the exhibition of the instrument with its axle either the greatest 

 or the least axis of inertia. The dimensions and weights of the parts of the top 

 which I have found most suitable, are given in a note at the end of this paper. 



Now let us make the axes of inertia in the plane of the ring unequal. We 

 may do this by screwing the balance screws x and x 1 farther from the axle 

 without altering the centre of gravity. 



Let us suppose the bob B screwed up so as to make the axle the axis of 

 least inertia. Then the mean axis is parallel to xx l , and the greatest is at right 

 angles to xx 1 in the horizontal plane. The path of the invariable axis on the 

 disc is no longer a circle but an ellipse, concentric with the disc, and having 

 its major axis parallel to the mean axis xx 1 . 



The smaller the difference between the moment of inertia about the axle and 

 about the mean axis, the more eccentric the ellipse will be; and if, by screwing 

 the bob down, the axle be made the mean axis, the path of the invariable axis 

 will be no longer a closed curve, but an hyperbola, so that it will depart alto- 

 gether from the neighbourhood of the axle. When the top is in this condition 



