332 Sir William Thomson on the Motion of 



inductive, and the part of its seolotropy (if any) which depends 

 on inductive magnetization being due either to magnecrystallic 

 quality of its substance or to its shape*. 



On the Motion of Rigid Solids in a Liquid circulating irrotation- 

 ally through perforations in them or in a Fixed Solid f, 



1. Let ty, <j>, . . . be the values at time t, of generalized coor- 

 dinates fully specifying the positions of any number of solids 

 movable through space occupied by a perfect liquid destitute of 

 rotational motion, and not acted on by any force which could 

 produce it. Some or all of these solids being perforated, let 

 %> x'j %"> ^ c - De tne qualities of liquid which from any era of 

 reckoning, up to the time t, have traversed the several apertures. 

 According to an extension of Lagrange's general equations of 

 motion, used in voL i. of Thomson and Tait's ' Natural Philo- 

 sophy/ §§331-336, proved in §§ 329,331 of the German 

 translation of that volume, and to be further developed in the 

 second English edition now in the press, we may use these quan- 

 tities %, X*> ' • • as *? ^ ne y were coordinates so far as concerns the 

 equations of motion. Thus, although the position of any part 

 of the fluid is not only not explicitly specified, but is actually 

 indeterminate, when ty, ip, . . . ^, %'*•*• are a ^ given, we may 

 regard ^, %j . . . as specifying all that it is necessary for us to 

 take into account regarding the motion of the liquid, in forming 

 the equations of motion of the solids; so that if f, 97, ... , and 

 M*, <£> . . . denote the generalized components of momentum and 

 of force [Thomson and Tait, § 313 (a) (£)] relatively to ty, <£..., 

 and if k, k\ ... K, K7 . . . denote corresponding elements rela- 

 tively to %, %', . . . , we have (Hamiltonian form of Lagrange's 

 general equations) 



d£ *t , T „ d v trT _ 



dt + <ty ' dt + d(j> '' 



d* ,OT_ & *T__ K , 



dt + dx--> dt + d x '-^>- 



(1) 



where T denotes the whole kinetic energy of the system, and ft 

 differentiation on the hypothesis of £, rj, . . . k, k' . . . constant. 



* "Theory of Magnetic Induction in crystalline and non-crystalline sub- 

 stances," Phil. Mag., March 1851 5 " Forces experienced by inductively 

 magnetized ferro-magnetic and diamagnetic non-crystalline substances," 

 Phil. Mag., Oct. 1850; " Reciprocal action of diamagnetic particles,' , 

 Phil. Mag., Dec. 1855; all to be found in a collection of reprinted and 

 newly written papers on electrostatics and magnetism, nearly ready for 

 publication (Macmillan, 1872). 



f The title and first part (§§ 1-13) are new; the remainder (§§ 14, 15) 

 was communicated to the Royal Society at the end of last December. — 

 W. T., September 26, 1872. 



