Perforated Solid in a Liquid, 495 



the momenta Kp, /c'p,. . . only. By means of the relations 



-r,^ r-^— + q=^=X, &c., &c. 



d-dV 3T' BT' 3T* • BT' ' . 



T - w^— + v^——r=r hg-^- =■!«, &c, &c. 



a/^dp o^ o^ o<? o^ 



the equations of motion of the solid can at once be written 

 down. X, : . ., L, . . ., are of course impressed force- and couple- 

 constituents. 



10. Since the kinetic energy due to any number of peiv 

 forated solids, moving in circulating liquid, can be divided 

 into two parts, of which one is a function of the component 

 velocities of the solids alone, and the other a function of the 

 circulation-momenta alone, the above method may obviously be 

 extended ; in fact a slight change in (14) will render it at 

 once applicable to the more general case. We shall have, 

 evidently, 



Y=E-K + tuttcp ff/z-^W + similar terms in v, w, 

 + %p%Kp\ I \ny— mz— W^V°" + similar terms in q, r, (15) 



where E is still the energy due to the motion of the solids and 

 the acyclic motion of the liquid, and K the energy due to the 

 circulations. In each barrier-term the first £ denotes sum- 

 mation with respect to all the solids, and for each u or p, &c, 

 the second 2 denotes summation with respect to all the barriers 

 of the system. 



These hydrodynamical results are not new, but the method 

 of proof is in some respects different from anything that has 

 yet been given, and will, I hope, be found intelligible and fairly 

 simple. In an admirable memoir, just communicated to the 

 Physical Society, Mr. Bryan has given a direct hydrodynamical 

 proof of the equations holding good for the motion of the 

 system in question ; but it seemed to me also desirable that 

 the problem should be rigorously treated by the method of 

 generalized coordinates, avoiding any assumption as to the 

 impulse of the cyclic motion, and proceeding entirely from the 

 principles established by Lagrange, and extended by Hamilton, 

 Routh, and Hayward. 



When this paper was in proof it contained some remarks 

 on the ignoration of coordinates, as treated in Thomson and 

 Tait's 'Natural Philosophy'*. Calling %,%',... tne i»de- 

 * Part I. § 319, example G. 



