SIR W. THOMSON ON VORTEX MOTION. 225 



the space bounded by S, because of our present condition, that no solid is cut by 

 S. Hence every statement and formula of § 15, as far as equation (11), may be 

 now applied to the matter within S ; but instead of (12) we now have [Thomson 

 & Tait, § 736], if we denote by T 15 T 2 , &c, another set of surface spherical 

 harmonics, 



* = *• + *!' + s ^' 2 + & M ri <n* 



+ T x r- 2 + T 2 r~ 5 + &c J K J ' 



for all space between the greatest and smallest spherical surface concentric with 

 S, and having no solids in it, because through all this space, § 16, and the equa- 

 tion of continuity prove that y 2 <p = 0. Hence, instead of (13), we now have 



R = ~ P = S x + 2r S 2 + 3r 2 S 8 , &c. 



(15). 



Hence finally 



dt = i JJ V *<!& ~ << + l) T < ^J sln mi + • < 16 >- 



Now if, as assumed in § 5, neither any moveable solids, nor any part of the 

 boundary exist within any finite distance of S all round ; S T , S 2 , &c, must each 



dlu 

 be infinitely small : and therefore (16) gives -jr = 0. This proves the proposition 



asserted in § 5 : because a system of forces cannot have zero moment round 

 every line drawn through any finite portion of space, without having force-resul- 

 tant and couple-resultant each equal to zero 



20. As the rigidity of the solids has not been taken into account, all or any of 

 them may be liquefied (§ 3) without violating the demonstration of § 19. To save 

 circumlocutions, I now define a vortex as a portion of fluid having any motion 

 that it could not acquire by fluid pressure transmitted through itself from its 

 boundary. Often, merely for brevity, I shall use the expression a body to denote 

 either a solid or a vortex, or a group of solids or vortices. 



21. The proposition thus proved may be now stated in terms of the definitions 

 of § 6, which were not used in § 5, and so becomes simply this: — The impulse of 

 the motion of a solid or group of solids or vortices and the surrounding liquid remains 

 constant as long as no disturbance is suffered from the influence of other solids or 

 vortices, or of the containing vessel. 



This implies, of course (§ 6), that the magnitudes of the force-resultant and 

 the rotational moment of the impulse remain constant, and the position of its axis 

 invariable. 



T 



* There is no term — , because this would give, in the integral of flow aci'oss the whole sphe- 

 rical surface, a finite amount of flow out of or into the space within, implying a generation or 

 destruction of matter. 



VOL. XXV. PART I. 3 M 



