124 ON THE MOTION OF SOLIDS THRO-UGH A LIQUID. [CHAP. V. 



When no external forces act, these equations have the first 

 integrals 



f + 77 2 +r 2 = const., 



^f + M + v ~ const., 



u% + VTJ + w% + p\ + q/j, + rv = 2T = const., 



of which the first and second together express the fact that the 

 magnitudes of the force- and couple-constituents of the impulse 

 are constant, and the third the fact that the whole energy of 

 the motion is constant. 



110. It remains to express f, rj, f, &c. in terms of u, v y w, &c. 

 In the first place let denote the kinetic energy of the fluid 

 alone, so that 



where the integration extends over the whole surface of the 

 moving solid. Substituting in this formula the value (2) of &amp;lt;, we 

 get for 2@ an expression of the form 



= An? + Bv* + Cw z + 2A vw + ZB wu + 2 C uv 

 + Pp* + Q&amp;lt;f + Rr* + 2P&amp;gt; + ZQ rp + 2Kpq 



+ Mv+ Nw) 



+M v+ N w} 



+ M&quot;v + N&quot;w) ................... ..... ..... (21), 



where A, B, C, &c. are certain constant coefficients whose values 

 depend only on the form of the solid, and on the position of the 

 axes of co-ordinates relative to the solid ; viz. we have 



mdS, 



^ ( 

 &o., &c., &c. 



.(22), 



