118-119] KINETIC ENERGY. 173 



with similar expressions for B, 0, Q, R. The remaining coefficients, as will 

 appear presently, in this case all vanish. We note that 



2(oo-0o) 



so that if a>b>c, then A<B<0, as might have been anticipated. 



The formulae for an ellipsoid of revolution may be deduced by putting 

 b=c; they may also be obtained independently by the method of Arts. 101- 

 106. Thus for a circular disk ( = 0, b = c) we have 



A=|pc 3 , B = C=0, } 



' (m). 



The kinetic energy, Tj say, of the solid alone is given by an 

 expression of the form 



2Tj = m (u* + v 2 + w*) 



+ Pip 2 + Qi? 2 + Rir 2 + ZPSqr + 2Q 1 'rp + 2R/pg 



+ 2m {p (0w -yv) + q (yu - aw) + r (av - flu)} ...... (4). 



Hence the total energy T + T lf of the system, which we shall 

 denote by T, is given by an expression of the same general form as 

 (2), say ' 



2T = Au* + Bv* + CV + ZA'vw + Wwu + Zffuv 



+ Pp* + Qq* + Rr* + ZP'qr + ZQ'rp + 2R'pq 

 + 2p (Lu + Mv + Nw) 



} .............................. (5), 



where the coefficients are printed in uniform type, although six of 

 them have of course the same values as in (4). 



119. The values of the several components of the impulse in 

 terms of the velocities u, v, w, p, q, r can now be found by a well- 

 known dynamical method *. Let a system of indefinitely great forces 

 (X, Y, Z, L, M, N) act for an indefinitely short time r on the solid, 

 so as to change the impulse from (f, 77, X, p, v) to 



AT;, ?+ A?, X + AX, /* + A/*, v + A*/). 



* See Thomson and Tait, Natural Philosophy, Art. 313, or Maxwell, Electricity 

 and Magnetism, Part iv. , c. v, 



