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XXXIY. A Hydro dynamical Proof of the Equations of Motion 

 of a Perforated Solid, with Applications to the Motion of a 

 Fine Rigid Framework in Circulating Liquid. By G. H. 

 Bryan*. 



Introduction. 



1. TN the whole range of hydrodynamics, there is probably 

 JL no investigation which presents so many difficulties 

 as that which deals with the equations of motion of a per- 

 forated solid in liquid. The object of the present paper is to 

 show how these equations may be deduced directly from the 

 pressure-equation of hydrodynamics, without having recourse 

 to the laborious method of ignoration of coordinates. The 

 possibility of doing this is mentioned by Prof. Lamb in his 

 'Treatise on the Motion of Fluids' (pp. 119, 120), but he 

 dismisses the method with the brief remark that in most cases 

 it would prove exceedingly tedious. I think, however, that 

 it will be admitted that the following investigation is more 

 straightforward and simple than that given by Basset in his 

 ' Hydrodynamics/ vol. i. pp. 167-178. 



The usual method presents little difficulty when the motion 

 of the liquid is acyclic, because the whole motion could in 

 such cases be set up from rest by suitable impulses applied to 

 the solids alone ; and a consideration of Routes modified 

 Lagrangian function shows that in this case the equations of 

 motion can be obtained by expressing the total kinetic energy 

 as a quadratic function of the velocity-components of the solid 

 alone, and applying the generalized equations of motion re- 

 ferred to moving axes. 



If, however, the solid is perforated, and the liquid is circu- 

 lating through the perforations, this method presents several 

 difficulties. If the solid were reduced to rest by the applica- 

 tion of suitable impulses, the liquid would still continue to 

 circulate through the perforations, the " circulation " in any 

 circuit remaining unaltered. From this and other circum- 

 stances we are led to infer that these circulations are not 

 generalized velocity-components, but rather that the quan- 

 tities Kp are generalized momenta. Now the kinetic energy 

 of the system is naturally calculated as a function of the 

 velocity-components of the solid and of these constant circu- 

 lations (or the corresponding momenta) ; a form unsuited for 

 obtaining the equations of motion. We ought either to have 

 the kinetic energy expressed in terms of generalized velocity- 

 components alone, or to know the "modified Lagrangian 



* Communicated by the Physical Society : read February 24 ; 1893. 



