SIB W. THOMSON ON VORTEX MOTION. 235 



or moving solids, and the other none. The resultant-momentum of the whole 

 matter enclosed by the first is in the direction of the impulse, and is equal to § 

 of its value. The resultant-momentum of the whole fluid enclosed by the 

 second is the same as if it all moved with the same velocity, and in the same 

 direction, as at its centre. 



III. Imagine any two infinite planes at a finite distance from one another 

 and from the field of motion, but neither cutting any solid or vortex. The com- 

 ponent perpendicular to them of the momentum of the matter occupying at any 

 instant the space between them (whether this includes none, some, or all of the 

 vortices or moving solids) is zero. 



46. To prove these propositions: — 



I. Consider in either case a finite length of the prism extending to a very 

 great distance in each direction from the field of motion, and terminated by 

 plane or curved ends. Then, the motion being, as we may suppose (§ 61) started 

 from rest by impulsive pressures on the solids [or (§ 66) on the portions of fluid 

 constituting the vortices] ; the impulsive fluid pressure on the cylindrical surface 

 can generate no momentum parallel to the length ; and to generate momentum 

 in this direction there will be, in case 1, the impressed impulsive forces on the 

 solids, and the impulsive fluid pressures on the ends ; but in case 2 there will 

 be only the impulsive fluid pressure on the ends. Now, the impulsive fluid 

 pressures on the ends diminish [§ 50 (15)] according to the inverse square of the 

 distance from the field of motion, when the prism is prolonged in each direction, 

 and are therefore infinitely small when the prisms are infinitely long each way. 

 Whence the proposition I. 



47. By using the harmonic expansions § 19, (14), (15), in the several expres- 

 sions (1), (2), of § 37, (1), (2); and the fundamental theorem 



of the harmonic analysis [Thomson & Tait, App. B. (16)]; and putting S* = 

 for one case, and T; = for the other ; we prove the two parts of Prop. II., § 45 

 immediately. 



48. To prove Prop. II., § 45, the well-known theory of electric images in a 

 plane conductor* may be conveniently referred to. It shows that if N x denotes 

 the normal component force at any point of an infinite plane due to any distribu- 

 tion, /", of matter in the space lying on one side of the plane, a distribution of 



matter over the plane having ^- N t for surface density at each point exerts the 



same force as m through all the space on the other side of the plane, and therefore 

 that the whole quantity of matter in that surface distribution is equal to the 



* Thomson, Camb. and Dub. Math. Journal, 1849; Liouville's Journal, 1845 and 1847; or 

 Reprints of Electrical Papers, (Macmillan, 1869.) 



