109 112.] KINETIC ENERGY. 125 



the transformations being effected by the use of (3) and of a 

 particular case of Green s theorem. These expressions for the 

 coefficients are due to Kirchhoff. 



The kinetic energy, W say, of the solid alone is also given by 

 a quadratic function of u, v, w, &c., in which however A, B, G are 

 each equal to the mass of the solid, whilst A , If, C , L, M, N, &c. 

 all vanish. The total energy ?& + *& (= T, say,) of the system is 

 therefore given by a formula of the same form as (21). Except 

 when otherwise indicated we shall suppose A, B, C, &c. to stand 

 for the coefficients in the expression for twice this total energy. 



111. The only form of solid for which the coefficients in 

 the expression (21) for 2^ have been actually determined is the 

 ellipsoid. \Ve readily find 



F 



where the notation is the same as in Art. 102, Ex. 3. The values 

 of B, C, Q, R may be written down from symmetry ; those of the 

 remaining coefficients are all zero. See Art. 116 (d). Since 



it appears that if a &amp;gt; b &amp;gt; c, then A &amp;lt; B &amp;lt; C, as might have been 

 anticipated. 



112. When in any dynamical system the expression for the 

 kinetic energy in terms of the velocities is known, the values of 

 the component momenta can be derived by a perfectly general 

 process. For this we must refer to books on general Dynamics*. 

 Applied to our case it gives 



dT dT dT dT dT dT 



respectively. These formulae are readily deduced from those 

 which relate to a perfectly free rigid body by supposing the 



* See Thomson and Tait, Nat. Phil. Art. 313, or Maxwell, Electricity and 

 Magnetism, Part iv. c. 5. An outline of the process adapted to our case is given in 

 note (C). 



