Motion of a Perforated Solid in Liquid. 345 



f = 2*,JJ(z-^)fc,&c (16) 



x=s^rr^-7^-^-w &c. . . .(17) 



As Lamb has pointed out (' Motion of Fluids/ p. 140), the 

 six quantities (f ? rj, £, X, //,, v) are " the components of the 

 impulse of the cyclic fluid motion which remains when the 

 solid is (by forces applied to it alone) brought to rest " *. They 

 are linear functions of the circulations and their form depends 

 on the form of the solid. If there is only one aperture they 

 are all proportional to the circulation k. 



The Modified Lagrangian Function. 



8. We shall now show that the motion of the solid can be 

 determined in terms of Eouth's modified Lagrangian function, 

 and shall find the form of this function for the system. 

 Putting 



H. = T + £u + V v + & + \p + f iq + vr + F{ f cp) ) . . (18) 



where F(/ep) denotes any function whatever of the quantities 

 Kp, we see that the equations of motion reduce to the standard 

 form 



at du ov ] ow 



T d^R 3H ^H 3H ^H 



L=^-^ w^ bv^— —r^ r-#^— • . (20) 



dt op ov ow oq or 



The function H, therefore, plays the same part in deter- 

 mining the equations of motion of the solid as the kinetic 

 energy T in the case of an imperforated solid (or any solid 

 when the motion of the liquid is acyclic). It remains (i.) to 

 determine what quantities are to be regarded as the generalized 

 velocities if the quantities Kp are regarded as generalized 

 momenta ; (ii.) to find the form of the function F(/ep) in order 

 that H may represent the modified Lagrangian function. 



9. Let y m be the generalized velocity-component corre- 

 sponding to the ignored momentum tc^p. Then, as Routh has 

 shown (' Rigid Dynamics,' vol. i. § 420), the modified La- 

 orano-ian function H is of the form 



H=T-2* m pXm, (21) 



* Our I, rj, £, A, jx, v are the same as the £ , rj , £ , X , p 0i v of Lamb, or 

 the £, g), 3, 8, m, Sfl of Basset's < Hydrodynamics.' 



