316 ON THE MOTION OF 



but different applications of this extensive branch of Demon- 

 strative Mechanics. What adds to the interest and value of 

 this subject is the circumstance that a large class of oscillatory 

 motions, namely those of any rigid system whatever whose 

 points depart but little from the position which they occupy 

 when at rest, has been found susceptible of complete deter- 

 mination, by means of which the position of the bodies com- 

 posing the system, may be expressed (to use the language of 

 analysis) in finite functions of the time. The general prob- 

 lem is one, however, of the greatest difficulty, and even ap- 

 proximate solutions can rarely be obtained except when the 

 conditions of the question restrict within near limits some of 

 the variations of the system. Every contribution, therefore, 

 however trifling, to this branch of analysis, is entitled to a 

 favourable reception, and it is this reflection which encourages 

 me to offer to the Society the fruits of an attentive considera- 

 tion of some portions of this subject. The memoir which I 

 have ventured to present to them is a general dissertation 

 upon the Dynamics of solids on supporting surfaces, in the 

 two hypotheses of perfect sliding and perfect rolling, with a 

 special consideration of the laws of their oscillatory motions. 

 The formulae which I have given, besides their use in a variety 

 of geometrical and mechanical speculations, conduct as it will 

 be found to a complete solution of the problem of the oscilla- 

 tions of a supported body of any form and law of density 

 whatever revolving on a plane or spherical surface with any 

 initial velocity compatible with small deviations of the natu- 

 ral vertical of the body from its position when at rest : sup- 

 posing either the absence of all friction or the action of a 

 friction which prevents all sliding motion, but which al- 

 lows the body, at the same time that it revolves round the 

 normal, to roll in all directions from the variable point of 

 contact. The same formulae will conduct to the solution of 

 a great variety of analogous problems, in which the excur- 

 sions of some part of the system are confined to the imme- 

 diate neighbourhood of its equilibrium position. They are 

 susceptible moreover of easy adaptation to any hypothesis of 



