«ni.ins OX si'RFACES. >T I 



cesser (Laeroix, Cut. Int. Vol. II. p. 37.) En the course 

 of this computation, into which the limits of the present com- 

 munication will not allow me to enter, equations of limitation 

 will arise showing the conditions of the oscillatory motions of 

 ilif body. These equations will in general be expressed in 

 the form of relations between the constants which determine 

 the form and magnitude of the areolas of contact, the magni- 

 tude ami density of the body and the position of its centre of 

 gravity. Among the oscillatory motions possible, there is one 

 of a peculiar nature which 1 do not recollect ever having seen 

 remarked, — I mean when the motion is around a state of equi- 

 librium, stable from the form of the moving body but unstable 

 from the form of the supporting surface, or the contrary; as 

 for example, when an ellipsoid is balanced on the outer surface 

 of a sphere, the summit of the shortest axis of the ellipsoid 

 lii-in;;- in contact with the highest point on the surface of the 

 sphere. Into such a position we may conceive the ellipsoid 

 to have descended from some assignable initial place of rest, 

 oi some combination of position and velocity. A motion 

 would ensue which in a variety of cases would he oscillatory. 

 The oscillations would however he liable to he broken by the 

 application of the slightesl I'orce. and would he followed by 

 the entire departure of the body from the place it occupied. 

 These motions may he called unstable list illtitinns. Tiny 

 hear the same relation to stabk oscillations that unstable does 

 to stable equilibrium. 



With respecl to the four linear equations above obtained, 

 I shall only add (hat in the present case they may he imme- 

 diately reduced by eliminating | and r to two equations of 

 tlii- fourth order of the form 



■I - H- n 7i7V+ C w -f- D ,, - / ,,, - III, = ... 



\ on. 111. "i I; 



