Solid Bodies through a Liquid. 333 



2. To illustrate the meaning of %, K, k, tf, , * . , let B be one 

 of the perforated solids, to be regarded generally as movable, 

 draw an immaterial barrier surface 12 across the aperture to 

 which they are related, and consider this barrier fixed rela- 

 tively to B. Let N denote the normal component velocity, 

 relatively to B and Q, of the fluid at any point of XI ; and let 

 f J dcr denote integration over the whole area of H ; then 



$$Nde=x, (2) 



and 



X-PIf 1 ^ (») 



which is a symbolical expression of the definition of p£. To the 

 surface of liquid coinciding with 12 at any instant, let pressure 

 be applied of constant value K per unit of area, over the whole 

 area ; and at the same time let force (or force and couple) be ap- 

 plied to B equal and opposite to the resultant of this pressure 

 supposed for a moment to act on a rigid material surface Q, 

 rigidly connected with B. The " motive " (that is to say, system 

 of forces) consisting of the pressure K on the fluid surface, and 

 force and couple B as just defined, constitutes the generalized 

 component force corresponding to % [Thomson and Tait, 

 § 313 (#)]; for it does no work upon any motion of B or other 

 bodies of the system if ^ is kept constant ; and if % varies, work 

 is done at the rate 



K% per unit of time, 



whatever other motions or forces there may be in the system. 

 Lastly, calling the density of the fluid unity, let k denote " cir- 

 culation"* [V. M. § 60 («)]t of the fluid in any circuit crossing 

 /9 once, and only once; it is this which constitutes the genera- 

 lized component momentum relatively to % [Thomson and Tait, 

 § 313 (e)]-, for (V.M. § 72) we have 



«=1K<#, (4) 



if the system given at rest (or in any state of motion for which 

 k — 0) be acted on by the motive K during time /J. 



* Or \Fds if F denote the tangential component of the absolute velo- 

 city of the fluid at any point of the circuit, and \ ds line integration once 



round the circuit. 



f References distinguished by the initials V. M. are to the part already 

 published of the author's paper on Vortex Motion (Transactions of the 

 Royal Society of Edinburgh, 1867-8 and 1868-9). 



, • f The general limitation, for impulsive action, that the displacements 

 effected during it are infinitely small, is not necessary in this case. Com- 

 pare § 5 (11) below. 



