105 107.] MOTION OF A SPHERE. 121 



very simple example of this method we may take the case of the 

 rectilinear motion of a sphere, which has been already investigated 

 otherwise. By the formula of Art. 65, the kinetic energy of the 

 fluid 



which in our case 



by (12), or finally \M F 2 . Hence the total energy of the system 

 is 



The rate at which this is increasing, i.e. 



must be equal to XV, the rate at which the impressed force X 

 does work. Discarding the common factor Fwe are led again to 

 the equation (19). 



107. In the general case the motion of the fluid at any 

 instant depends, as we saw in Art. 100, only on the values of the 

 quantities M, v, w, p, q, r used to express the motion of the solid ; 

 so that the whole dynamical system is virtually one of six degrees of 

 freedom, although it differs in some respects from the kind of system 

 ordinarily contemplated in Dynamics. Thomson* and Kirchhoff*f 

 have independently shewn how a system of the peculiar kind here 

 considered may be brought under the application of the ordinary 

 methods of that science. We shall, in what follows, adopt Thom 

 son s procedure, with some modifications. 



Whatever be the motion of the fluid and solid at any instant, 

 we may suppose it produced instantaneously from rest by the 



* Phil. Hag. November, 1871. 



+ Crelle, t. 71. See also Vorlesungen uber Math. Physik. Mechanik. c. 10. 



