172 



MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



118. It remains to express f, rj, f, X, /A, v in terms of 

 u, v, w y p, q, r. In the first place let T denote the kinetic energy 

 of the fluid, so that 



2T = - 



^-dS. 



.(1), 



where the integration extends over the surface of the moving 

 solid. Substituting the value of <, from Art. 115 (2), we get 



2T = Ait 2 + "Bv- + Cw 2 + 2A.'vw -* 2BW + 2C'uv 



+ 2j9 (LM 

 + 2q (Ii'u 



Rr 2 + 2P'qr + 2Q'rp + 2R'pq 

 + Nw) 

 't> + N'w) 



(2), 



where the 21 coefficients A, B, C, &c. are certain constants 

 determined by the form and position of the surface relative to the 

 coordinate axes. Thus, for example, 



= p II (f>. 2 ndS = p II 



p = ~ 



ds = 



(3), 



the transformations depending on Art. 115 (3) and on a particular 

 case of Green's Theorem (Art. 44 (2)). These expressions for 

 the coefficients were given by Kirchhoff. 



The actual values of the coefficients in the expression for 2T have been 

 found in the preceding chapter for the case of the ellipsoid, viz. we have 

 from Arts. Ill, 112 



