Perforated Solid in a Liquid. 493 



ignore all other coordinates, and from the kinetic energy 



expressed as a function of 6, 6', . . . %, %',... WI "ite down the 

 Lagrangian equations for 0,6', . . . and, if we wish, for^,^, . . . 

 also. These latter, however, are less directly intelligible, 

 since in general they involve finite pressures continuously 

 acting over geometrical surfaces drawn through the liquid. 



8. If we wish to picture the application of the principle of 

 least action (§ 3) to the present case, we may proceed as 

 follows: — Let the system start from the configuration (I.) 

 and move without additional constraint or influence to the 

 configuration (II.)- Then let it start again from the confi- 

 guration (1.) with the same velocities as before, and during 

 the motion let infinitesimal additional forces act on the solid, 

 while infinitesimal pressures, uniform over each barrier- 

 surface, are impressed on the liquid ; the total rate at which 

 the additional influences do work being at each instant zero. 

 Further, let the additional influences be so adjusted that the 

 system, after following a slightly different path, passes through 

 a configuration such that 6, &, . . . %, %'>••• are all the same 

 as for (II.). Then, to pass from the configuration (II.) to the 

 present configuration requires no displacement of the solid, 

 and only such displacement of the liquid that the total volume 

 which crosses any barrier-surface is zero. In such a change 

 of configuration impulses of the types 6, 6', . . . % %')••• would 

 have no " virtual moment," just as forces applied to the solid 

 and uniform pressures applied to the barrier-surfaces would 

 give rise to no virtual work. 



9. At this stage it will be convenient to replace 6, 6 1 , . . . 

 by the components u, v, w of linear velocity and p, q, r of 

 angular velocity, which determine the instantaneous motion 

 of the solid along and about axes fixed in itself. The Lagran- 

 gian equations for the six coordinates 6, 6 1 ,... must accord- 

 ingly be replaced by the forms suitable to moving axes. The 

 expression for the energy in terms of the velocities now 

 becomes a homogeneous quadratic function of u, v,- w, p, q, r, 



% %',... in which all the coefficients are known to be 

 constants. 



Let us apply the method due to Routh *, and modify this 

 function with respect to the coordinates ^, %',... If T be the 

 value of the kinetic energy in terms of the velocities alone, the 

 modified function («'. e. the kinetic part of Routh's modified 

 Lagrangian function) 



T=T -^-%^~ •••=t-<vx-«W- ■ .(7) 



* ' Rigid Dynamics/ vol. i. chap. viii. 



