Equations in certain Cases of Fluid-Motion. 359 



Again ; 



^T ^T ^T 



du dv dw 



are the force-components of the impulse with respect to O'X', 

 O'Y', O'Z' ; 



du dv dw 

 d-yjr dyjr dty 



correspond to the displacement of a vector u, v, w fixed in 

 space with respect to the moving axes, owing to the motion dyjr 

 of the latter ; 



[vector u, v, w 



dT _ p component along R of relative displacement of 



dty ' dyfr 



= — moment of a force E applied at end of vector 

 u, v 3 iv round axis O x Z y ; 



.*. Lagrange's equation 



dt dyjr dtfr "" 

 means that 



j- (moment of impulse round O r Z ; ) 



+ moment of force of impulse applied at end of vector 

 u, v, w round the same line = 0. 



The left-hand member is obviously 



= -=- (moment of impulse round OZ) ; 



.*. this last moment is constant, and the moments round OX 

 and OY are also constant. 



Secondly, let there be more rigid bodies than one. We 

 can now assume as the generalized coordinates the same as 

 we have just taken which have reference to one of the bodies, 

 together with other coordinates depending entirely upon the 

 relative position of the rigid bodies amongst themselves ; then 



■jrn 7m 



■Y-. and — - are evidently the force-components along the axis 

 dx c lty 



OX and the moment round the axis 0% of the whole impulse, 



and the reasoning runs as before. 



