232 SIR W. THOMSON ON VORTEX MOTION. 



forces (§ 3) were applied, or determinate distributions in infinitely thin layers 

 at their surfaces, may be found, which through all the space external to them 

 shall produce the same potential as the velocity-potential, and therefore the same 

 distribution of force as the distribution of velocity through the whole fluid. 

 But inasmuch as when the magnetic force in the interior of a magnet is 

 defined in the manner explained in §48(2) of my "Mathematical Theory of 

 Magnetism, 1 '* it is expressible through all space by the differential coefficients of 

 a potential ; and, on the contrary, for the kinetic system u dx + v dy + w dz is 

 not a complete differential generally through the spaces occupied by the solids, 

 the agreement between resultant force and resultant flow holds only through the 

 space exterior to the magnets and solids in the magnetic and kinetic systems 

 respectively. But if the other definition of resultant force within a magnet, 

 ["Math. Theory of Magnetism," § 77, foot-note, and § 78], published in preparation 

 for a 6th chapter " On Electro-magnets" (still in my hands in manuscript, not 

 quite completed), and which alone can be adopted for spaces occupied by non-mag- 

 netic matter traversed by electric currents, the magnetic force has not a potential 

 within such spaces ; and we shall see (§ 68) that determinate distributions of 

 closed electric currents through spaces corresponding to the solids of the hydro- 

 kinetic system can be found which shall give for every point of space, whether 

 traversed by electric currents or not, a resultant magnetic force, agreeing in 

 magnitude and direction with the velocity, whether of solid or fluid, at the cor- 

 responding point of the hydrokinetic system. This thorough agreement for all 

 space renders the electro-magnetic analogue preferable to the magnetic; and, 

 having begun with the magnetic analogous system only because of its convenience 

 for the demonstration of § 38, we shall henceforth chiefly use the purely electro- 

 magnetic analogue. 



40. To prove the formula used in anticipation, in § 37 (1) we must now 

 (§§ 41, 42, 43) find the momentum of the whole matter — fluid, fluid and solid, 

 or even solid alone — at any instant within a closed surface S, in terms of the 

 normal component velocity of the matter at any point of this surface, or, which is 

 the same, the normal velocity of this surface itself, if we suppose it to vary so 

 as to enclose always the same matter. 



41. Let V be the volume of the space bounded by any varying closed surface 

 S. As yet we need not suppose V constant. Let x, y, z be the co-ordinates of 

 of the centre of gravity. We have 



Yx = Iff^dydz] .... (5), 

 where [ ] indicates that the expression within it is to be taken between proper 

 limits for S. Now as S varies with the time, the area through \\h\chffdy dz is 

 taken will in general vary ; but the increments or decrements which it experiences 



* Trans. R. S. Lond., 1851 ; or " Thomson's Electrical Papers." Macmillan. 1869. 



