332 ON THE MOTION OF 



a solution as it is possible to proceed in the present state of 

 the Calculus. The case however in which the natural vertical 

 of the body makes infinitely small conical oscillations around its 

 resting place, while the body itself revolves about this axis 

 with any velocity compatible with such oscillations, is com- 

 pletely solved by means of an analysis remarkable for its bril- 

 liancy, generality and rigour. The problem, it is shown, nat- 

 urally divides itself into two distinct portions, one in which 

 the form and density of the body is absolutely arbitrary, but 

 the rotation round the vertical small and consequently varia- 

 ble ; the other in which the rotation round the vertical is ar- 

 bitrary and consequently constant, but the form and density 

 of the body such that the conditions requisite to constitute 

 the natural vertical a natural axis of rotation shall be nearly, 

 though it is not necessary that they should be exactly, fulfil- 

 led. 



Poisson published his excellent Traite de Meccmique in 

 1811. In the second volume of this work, the author applies 

 his calculus to a determination of the motions of a homoge- 

 neous ellipsoid upon an inclined plane, both surfaces being 

 supposed perfectly smooth. The investigation does not bring 

 the formulas within the reach of the method of quadratures, 

 and therefore the problem cannot as yet be considered as 

 solved 47 . The author then proceeds to give an improved so- 

 lution of the question considered long before by Euler and 

 D'Alembert, of the motion of a solid body when it is sus- 

 tained upon a plane by a point fixed in the body, but mov- 

 ing freely along the plane. In the case in which the density 

 and figure are symmetrical about the axis joining the cen- 

 tre of gravity and sustaining point, the problem is reduced to 

 the method of quadratures, and a complete solution is given 

 in the hypothesis of small departures of the axis from some 

 intermediate inclination to the plane. In this solution Pois- 

 son has been followed by Prony in his Legons de Mecanique 



* 7 This reduction, it ought to have been remarked, is easily effected when the 

 ellipsoid becomes a spheroid of revolution. 



