-.ui.li)> on Bunp wes. 165 



bolic paraboloids. In cases of this kind, il may be well to 

 observe at once, the analysis does nol necessarily regard the 

 motion ronnd the normal as arrested by the impenetrability 

 of the rings, but implies in general a mutual penetrability so 

 as to admit but a single poinl of contact 



The law of continuity, a law to which analysis, in all its 

 processes, adheres with singular ami sometimes indeed with 

 inconvenienl faithfulness, requires us to attribute to both sides 

 of the supporting surface the power of feeling and sustaining 

 in both directions, the presence of the moving body. Thus, if 

 we suppose a sphere in motion on the outside of another 

 sphere, it would evidently einne. at some determinate epoch, 

 into a position where its pressure on the supporting surface 

 Would he nought. It would there leave the surface, and it- 

 motion afterwards would he a separate problem. An analy- 

 tical solution of the question however would regard the mov- 

 ing body as still connected with the surface of support, and 

 exerting on it a pressure tending to draw it outward from its 

 centre. This pressure would he such as would arise from a 

 momentary hut continually renewed connecting; thread infin- 

 itely short passing from sphere to sphere at the point of va- 

 riable contact, or such as would take place if we supposed the 

 surfaces of one () f the spheres to consist of two concentric 

 spherical surfaces infinitely near each other, and the momen- 

 tary poin! of contact of ihe other sphere to be always engaged 

 and confined between them. Again, let u< suppose that a 

 circle rolls and slides inside down an ellipsis whose maximum 

 curvature is greater and whose minimum is less than the 

 curvature of the circle. If we suppose moreover the long 

 axis vertical and the short axis longer than the diameter of 

 the circle, the circle in descending will come first to a plac< 

 where it will touch the ellipsis in two points and there phys- 

 ically it would stop, hut tlie analysis (on the hypothesis o| 

 one original poinl of contact) will consider the circle as geo- 

 metrical except at this point of contact, and of course will 

 represent the circle as passing onward unimpeded by this second 

 contact. It will then reach a point in the ellipse where the 



