222 SIR W. THOMSON ON VORTEX MOTION. 



Tu=fffdxdydz(imj-vz). . . . (5), 



ff denoting integration throughout this space (not now infinitesimal). But by (4) 



y w - m = \yjz- z dy)^ = d^ ■ (6); 



d 

 if -tt denote differentiation with reference to \^, in the system of co-ordinate 



x, p, ■vj', such that 



y = s cos -4/ , z = g sin ^ . . , . (7). 



Hence, transforming (5) to this system of co-ordinates, we have 



-L=fffdxd sg dl^ (8). 



Now, as the whole space is spherical, with the origin of co-ordinates in its centre, 

 we may divide it into infinitesimal circular rings with OX for axis, having each 

 for normal section an infinitesimal rectangle with dx and dp for sides. Inte- 

 grating first through one of these rings, we have 



dip 



dxdgs f 27r d% d *' 



which vanishes, because <p is a single- valued function of the co-ordinates. Hence 

 L = 0, which proves Lemma (2.). 



15. Returning now to the dynamical proposition, stated at the conclusion of 

 § 11 ; for the promised proof, let R denote the radial component velocity of the fluid 

 across any element, d<r, of the spherical surface, situated at (x, y, z) ; and let 

 u, v, w be the three components of the resultant velocity at this point ; so that 



B = «2 + „£-+«,* (9). 



4* -O* 4* 



The volume of fluid leaving the hollow spherical space across da-'m an infinitesimal 

 time, dt is TLdtr . dt, and the moment of momentum of this moving mass round 

 the centre has, for component round OX, 



(ivy — vz) Jicfo dt. 



Hence, if L denote the component of the moment of momentum of the whole, 

 mass within the spherical surface at any instant, t, we have (§ 11), 



Jt=lf( wy - vz)lid6 > ' • ■ ■ (10) - 



Now, using Lemma (1.) of § 12, and the notation of § 14, we have 



dip 



