216 



MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



and of C, C ,..., respectively. Hence (8) may be written in the 

 form 



where j3 lf /3 2 , ... are linear functions of C, C , ..., say 



.(12). 



The meaning of the coefficients a 1) 2 &amp;gt; , a i&amp;gt; s &amp;gt; &amp;gt; a PP ears fr m 

 the last line of (5), viz. we have 



dK 



dK 



.(13). 



Compare Art. 137 (18). 



If we now substitute from (11) in the equations (6) we obtain 

 the general equations of motion of a gyrostatic system/ in the form 



d 

 dt 



d 

 dt dq 



d^d 

 dt dq 3 



dq l 



T*^-^+X2,l)* 



dq 3 



(I,2)g 2 +(l, 



+ (2, 

 (3,2)&amp;lt;7 2 



dK 



dK 

 dq 2 



dK 



where 



dq. 



(14)*, 



.(15). 



The equations (21) of Art. 137 are a particular case of these. To complete 

 the identification it remains to shew that, in the hydrodynamical application, 



C= P K, C = P K ,..,, (i). 



For this purpose we may imagine that in the instantaneous generation of 

 the actual motion from rest, the positions of the various barriers are 

 occupied for a moment by membranes to which uniform impulsive pressures 

 pK, pKj . . . are applied as in Art. 54, whilst impulsive forces are simultaneously 

 applied to the respective solids, whose force and couple resultants are 

 equal and opposite to those of the pressures f. In this way we obtain a 

 system of generalized components of impulsive force, corresponding to the 



* These equations were obtained, in a different manner, by Thomson and Tait, 

 ?. c. ante p. 214. 



t Sir W. Thomson, I c. ante p. 211. 



