SOLIUM OB *i Hi \( B8, i I 9 



without sliding (the pura provoluiio of Leibnitz") will result, 



it is trui . for small motions from the usual hypotheses of 

 friction, but without some condition of this kind tin- body 

 would slip or slide as well as rock. Enter is the first who 



made this remark in the seventh volume of the Com- 

 mentaries of St Petersburg (1740), where he gives an 

 improved solution" of Bernoulli's problem, hut does not 

 appear to have been able, at that time. t<> determine what 

 would take place if the body were left free In slide as 

 well as to roll. Euler acknowledged his embarrassment 

 to D'Alembert in a letter to him dated 1746, and it is to 

 tin latter mathematician that we owe the first mhti^- 

 ful investigation of the problem when the surfaces in con- 

 tact are polished to a perfect smoothness. This solution is 

 given by D'Alembert in the second edition of his TraiU (/< 

 Dynamique, published in 1758, and is offered by him as an 

 instance of the utility of his now celebrated principle". His me- 

 thod is then applied to the case in which the horizontal plain- 

 opposes, by its roughness, a given degree of resistance to the 

 sliding motion, but the oscillations are still only of the kind 

 in which the axis of rotation retains throughout the mot ion 

 its original direction. This is a condition, however, which 

 restricts the problem to a case comparatively simple, for it i- 

 manifest that in general the axis of rotation will change con- 

 tinually its position in space, and the body must lie consider 

 ed as subject, Dot only to roll from side to side, but also to 

 pitch backward and forward, and at the same time to whirl 

 around the perpendicular drawn to the sin-face at the point 

 of contact. Hut before the triple rotation of a supported body 

 could be determined, it was necessary to investigate the phe- 

 nomena of the rotation of a free body, to which constrained 



G. d. I.. I »<■ linefi- super lines inccssu, ejuBque trilm.s gpecieb 

 radente, motu provolutionis, et composite ex ambobus. Jan. 1706. Act. Km. I 

 Lips. 1706, i'. I 1 '. 



De minimi o libus corporum tarn rigidorum quam flexibilium, me- 



thoda nova ac racilis. ( ora. \ id Petrop. 1740, p. 108. 



" Des Corps qui vacillant sur des plans. Traiti de Dynamique, I796,p 181 

 \ in,, in. — 4 M 



