SOLIDS ON SURFACES. 5 39 



readily to two integrations with respect to time, thus giving 

 .u mice the principles of the centre of gravity, of areas and of 

 living forces: but does not allow of integration with respect 

 to the dimensions of the system, without which ii is obvious 

 that the phenomena of its motion cannot in general he ascer- 

 tained. For this purpose either the equations of motion ob- 

 tained by this transformation must be employed to produce 

 six others which admit of this integration, as Laplace has done, 

 or these six must be obtained directly from the general dyna- 

 mical •equation by the application of the formulas involving the 

 variations of the angles of rotation round the axes of the body, 

 a method which was first carried fully into effect by Lagrange. 

 In his Mfcamque Analytique, he effects this transformation, 

 not by a direct method, but by means of his favourite subsid- 

 iary formulas. {Vol. I. p. 13, edit. 1811); and in doing so he is 

 under the necessity of warning the reader that the usual inter- 

 change of the differential of the variation and the variation of 

 the differential would not be legitimate with respect to the 

 quantities ddP, o<!<{, bdR. The difference arising from the 

 order in which the signs are placed (a difference obviously to 

 be ascribed to the incompleteness of both the differential and 

 th'' variation of the indefinite integrals P, Q, /?.) Lagrang< 

 then carefully investigates and takes into account. In a note 

 found among his papers after his death, and inserted in an 

 appendix at the end of the second volume of his Mtcanique, 

 he carries to its results, bya direct process, the Last mentioned 

 plan of transformation, and extends his analysis to all possible 

 systems, whether solid or not, thereby having regard to the 

 intestine or proper motions of the particles. As the method 

 indicated in this note appears to me to conduct to the neces- 

 sary results from the simplest principles, by the directest 

 means, and with the smallest quantity of analysis compatible 

 with a process entirely analytical, I shall devote a page or two 

 of this paper to the purpose of obtaining, by means of this 

 transformation, formulas preparatory to the solution of tin 

 problem I have proposed. On this subject I think it proper 

 to premise, that as the whole notation 1 have adopted refers 



s on. in. — 1 n 



