220 SIR W. THOMSON ON VORTEX MOTION. 



the momentum, and the moment of momentum, of the whole contents of the 

 vessel, let the vessel be spherical. Its impulsive pressure on the liquid will 

 always be reducible to a single resultant in a line through its centre, which (§ 4) 

 will be equal and contrary to the force-resultant of " the impulse ;" and, therefore, 

 with it will constitute in general a couple. The resultant, of this couple and the 

 couple -resultant of the impulse, will be equal to the moment of momentum of the 

 whole motion round the centre of the sphere (which is the centre of inertia). But 

 if the vessel be infinitely large, and infinitely distant all round from the moveable 

 solids, the moment of momentum of the whole motion is irrelevant; and what 

 is essentially important, is the impulse and its force and couple-resultants, as 

 defined above. 



9. The following way of stating (§§ 10, 12), and proving (§§ 11 — 15), a funda- 

 mental proposition in fluid motion will be useful to us for the theory of the 

 impulse, whether of the moveable solids we have hitherto considered or of vortices. 



10. The moment of momentum of every spherical portion of a liquid mass in 

 motion, relatively to the centre of the sphere, is always zero, if it is so at any one 

 instant for every spherical portion of the same mass. 



11. To prove this, it is first to be remarked, that the moment of momentum 

 of that part of the liquid which at any instant occupies a certain fixed spherical 

 space can experience no change, at that instant (or its rate of change vanishes at 

 that instant), because the fluid pressure on it (§ 1), being perpendicular to its 

 surface, is everywhere precisely towards its centre. Hence, if the moment of 

 momentum of the matter in the fixed spherical space varies, it must be by the 

 moment of momentum of the matter which enters it not balancing exactly that of 

 the matter which leaves it. We shall see later (§§ 20, 17, 18) that this balancing 

 is vitiated by the entry of either a moving solid, or of some of the liquid, if any 

 there is, of which spherical portions possess moment of momentum, into the fixed 

 spherical space; but it is perfect under the condition of § 10, as will be proved 

 in § 15. 



12. First, I shall prove the following purely mathematical lemmas; using the 

 ordinary notation u, v, w for the components of fluid velocity at any point 



(-», y> -)• 



Lemma (1.) The condition (last clause) of § 10 requires that udx + vdy + ndz 

 be a complete differential,* at whatever instant and through whatever part of the 

 fluid the condition holds. 



Lemma (2.) If udx + vdy 4- w dz be a complete differential of a single valued 

 function of oc, y, z, through any finite space of the fluid, at any instant, the con- 

 dition of ^ 10 holds through that space at that instant. 



* This proposition was, I believe, first proved by Stokes in his paper " On the Friction of 

 Fluids in Motion, and the Equilibrium and Motion of Elastic Solids." — " Cambridge Philosophical 

 Transactions," 14th April 1845. 



