236 SIR W. THOMSON ON VORTEX MOTION. 



whole quantity of matter in n* Hence, ff dor, denoting integration over the 



infinite plane 



ffXid*=Q (12). 



if the whole quantity of matter in m be zero. Hence, if N be the normal force 

 due to matter through space on both sides of the plane, provided the whole quan- 

 tity of matter on each side separately is zero, 



#N<Z«r=0 . (14); 



since N is the sum of two parts, for each of which separately (12) holds. This 

 translated into hydrokinetics, shows that the whole flow of matter across any 

 infinite plane is zero at every instant when it cuts no solids or vortices. Hence, 

 and from the uniformity of density which (§ 3), we assume, the centre of gravity 

 of the matter between any two infinite fixed parallel planes, has no motion in 

 the direction perpendicular to them at any time when no vortex or moving solid 

 is cut by either : which is Prop. III. of § 4 in other words. 



49. The integral flow of matter across any surface whatever, imagined to 

 divide the whole volume of the finite fixed containing vessel of § 1 into two parts is 

 necessarily zero, because of the uniformity of density ; and therefore the momen- 

 tum of all the matter bounded by two parallel planes, extending to the inner 

 surface of the containing vessel, and the portion of this surface intercepted 

 between them has always zero for its component perpendicular to these planes, 

 whether or not moving solids or vortices are cut by either or both these planes. 

 But it is remarkable that when any moving solid or vortex is cut by a plane, the 

 integral flow of matter across this plane (if the containing vessel is infinitely 

 distant on all sides from the field of motion), converges to a generally finite value, 

 as the plane is extended to very great distances all round from the field of 

 motion, which are still infinitely small in comparison with the distances to the 

 containing vessel ; and diminishes from that finite value to zero by another con- 

 vergence, when the distances to which the plane is extended all round begin to 

 be comparable with, and ultimately become equal to, the distances of the curve 

 in which it cuts the containing vessel. Hence we see how it is that the condition 

 of neither plane cutting any moving solid or vortex is necessary to allow § 46, 

 III. to be stated without reference to the containing vessel, and are reminded that 



* This is verified synthetically with ease, by direct integrations showing (whether by Cartesian 

 or polar plane co-ordinates), that 



r" r™ adydz 



d 



And taking -=- of this, we have 



rr i£+*=¥%*. * .... (12,-. 



the synthesis of (12). 



