SOLIDS ON SURFACES. 33;j 



Jlnalytique", Whewell in his Dynamics 49 , and various othei 

 authors and compilers. 



It is, I think, a matter of surprize, that oone of the Euro- 

 pean mathematicians should have thought of ascertaining 

 whether the method of Lagrange might not be successfully 

 employed in determining the variable pirouettes or oscillations 

 which a heavy body bounded by a giv< u surface will make on 

 a given plane or in general on any given surface of support. 

 The first solutions 1 have been able to find of any case what- 

 ever of this interesting question are contained in the eighth 

 number of the New York Mathematical Diary for July 

 isj7. The problem as proposed by Mr E. Nulty, of Phila- 

 delphia, requires a determination of all the small oscillations 

 which can be made by the segment of a sphere in contact 

 with a horizontal plane. Euler, as we have seen, had per- 

 fectly resolved this case, in the two hypotheses of perfect 

 sliding and perfect rolling, as long as the motion of rotation 

 is around an axis of invariable direction. lint the motion 

 round a variable axis he had carefully excluded, expressly on 

 the ground of its being inaccessible to the analysis of the 

 day. One of the solutions published in the work which I 

 have just mentioned is by I)r Admin, at that time Professoi 

 of Mathematics in Rutger's College, New Jersey. This so- 

 lution, which regards the segment as symmetrica] and mov- 

 ing without friction, begins with a verj ingenious and direct 

 transformation of Lagrang *s general formula of Dynamics 

 into another in which three of the variations are. a- usual, 

 variations of the coordinates of the centre of rotation, and the 

 other three, variations of the finite angles employ i I by Euler 

 and Laplace; a process which, though the most direct, has 

 not. a> far as 1 can ascertain, been pursued <>r even suggested 

 by any other author. The facility with which this problem, 

 as long a> friction is nol concerned, may be 1 subjected to the 

 methods and formulas of Lagrange, enabled me, in a solution 



48 Leeons de Mlcanique Analytique, Tome II. 1816, p 



45 A Treatise on Dynamics. 182a. p 



