122 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



action of a properly chosen set of impulsive forces applied to the 

 solid. This set, when reduced after the manner of Poinsot to a 

 force and a couple whose axis is parallel to the line of action of 

 the force, constitute what Thomson calls the impulse of the motion 

 at the instant under consideration. We proceed to shew that 

 when no external impressed forces act the impulse is constant in 

 every respect throughout the motion. 



108. The moment of momentum of a spherical portion of the 

 fluid about any line through its centre is zero; for this portion 

 may be conceived as made up of circular rings of infinitely small 

 section having this line as a common axis, and the circulation in 

 each such ring is zero. 



In the same way the moment of momentum of a portion of 

 the fluid bounded by two spherical surfaces about the line joining 

 the centres is zero. 



The moment of the impulse at any instant about any line is 

 equal to the corresponding moment of momentum at that instant 

 of the whole matter contained within a spherical surface having 

 its centre in that line and enclosing the moving solid ; for if we 

 suppose the motion generated instantaneously from rest, the only 

 forces which, besides those constituting the impulse, act on the 

 mass in question are the impulsive pressures on the spherical 

 boundary. Since these act in lines through the centre, they do 

 not affect the moment of momentum. 



It is, as was pointed out in Art. 99, immaterial whether we 

 simply suppose the fluid to extend to infinity and to be at rest 

 there, or whether we suppose it contained in an infinitely large 

 fixed rigid vessel which is infinitely distant in all directions from 

 the moving solid. The motion of the fluid within a finite distance 

 of the solid, and therefore the forces exerted by it on the latter, 

 are the same in the two cases. If we now suppose the infinite 

 containing vessel to be spherical in shape, and to have its centre 

 at any point P within a finite distance of the solid, the moment 

 of momentum of the included mass about any line through P is, 

 as we have just seen, equal to the moment of the impulse about 

 the same line. The same reasoning shews that if there be no 



