326 ON THE MOTION OF 



must have acquired in the highly probable hypothesis of its 

 original fluidity 35 . 



After the problem of free rotation had been solved, nearer 

 approaches were made to the determination of the motion of 

 a supported body. D'Alembert, who had briefly given in 

 the first volume of his Opuscules the modifications of his ge- 

 neral formulas applicable to this case, resumed the inquiry in 

 the fifth volume of the same work 36 . For this purpose he 

 undertakes a general solution of the question already consid- 

 ered by Euler. A body is supposed to be sustained by one 

 of its points upon a plane, and the circumstances of the mo- 

 tion are required. The resulting differential equations are, 

 however, so involved, that the author evidently abandons in 

 despair all idea of obtaining the necessary integrations. A 

 variety of simplifications and restrictions are then introduced 

 with a view to obtain cases admitting of first integrals. The 

 line which joins the centre of gravity and the point of sup- 

 port is supposed to be a principal axis, and the point is sup- 

 posed to move without friction on a horizontal plane, the 

 mode of considering* the resistances of friction and the incli- 

 nation of the plane being nevertheless laid down though found 

 to lead to unmanageable results. On the whole, D'Alembert 

 is far from having solved any but the simplest cases of this 

 problem, though he appears to have proceeded somewhat far- 

 ther than any of his cotemporaries. 



Euler, who had in the earlier volumes of the Commentaries 

 of the St Petersburg Academy considered, in conjunction 

 with Daniel Bernoulli, the effects of friction in retarding the 

 motion of polyhedral solids and homogeneous cylinders on 

 inclined planes 37 , turned his attention a few years before his 

 death to some varieties of the general problem of greater dif- 

 ficulty than these. His first memoir on this subject is divi- 



35 Thdorie <lc la librationde la Lune. Nouv. Mem. Berl. 1780. 



50 Sur le mouvement des Corps qui tournent. Opusc. Tome V. 1708. p. 489. 



37 De descensu corporum super piano inclinalo. — De nioiu corporum super 

 piano horizontal aspero. Com. Acad. Petrop. Tom. XIII. 1751. — De frictione 

 corporum rolantiuin. Novi Com. Acad. Petr. Tom. VI. 17G1. 



