366 ON THE 3IOTION OF 



curvatures of the two curves are equal, and where on one 

 side of the point of osculation the circle passes inside, and on 

 the other outside of the ellipsis. Before the circle comes 

 into this position the arc of contact is entirely within, after it 

 leaves it entirely without, the ellipsis, and the connection 

 must be maintained as in the preceding example. The same 

 remarks will apply to the motion of an ellipsoid placed within 

 a sphere of a curvature intermediate between the greatest and 

 least curvature of the ellipsoid, to all contacts between dis- 

 curvate surfaces, and in general to all cases in which the max- 

 imum curvature of one of the surfaces is not less than the 

 minimum curvature of the other. 



In order to determine the actual oscillatory motions of such 

 bodies, we must institute as many equations of condition 

 similar to (22) as the moving body can have points of contact 

 with the supporting surface. We must then determine 

 when the pressure at any one of these points becomes equal 

 to nought, after which the problem is to be considered as a new 

 one, and the subsequent motion of the body must be traced 

 by applying to it the equations resulting from one contact 

 less than before, until the body either again comes into a 

 fresh point of contact, or loses another of the contacts which 

 it was supposed to have at first. In the course of the various 

 positions into which the moving body would come, it would 

 frequently happen that two of the points would unite into 

 one by an inosculation of the curves of contact, or one would 

 become two, as when a sphere moves upon an oval annulus of 

 smaller dimensions than the sphere from the eoncurvate to the 

 discurvate portion of it. An inquiry into motions of this 

 kind is however foreign to the purpose of this paper, and I 

 return to the consideration of the problem when restricted to 

 a single point of contact. 



The selection of a paraboloid, in its three varieties of ellip- 

 tical, hyperbolical and intermediate, to serve as the osculating 

 figure of the areola at the balancing point of the body, is at- 

 tended with the advantage that, beside suiting all possible eases 

 of curvature, it is always applicable, whether the centre of 



