230 SIR W. THOMSON ON VORTEX MOTION. 



Professor Tait's plan, in such directions as to pass very near a convex fixed sur- 

 face. An ordinary 12-inch globe, taken off its bearings and hung by a thin cord, 

 answers very well for the fixed body. 



35. The investigation of §§ 30, 31, 32, is clearly applicable to a vortex or a 

 moving body, or of a group of vortices or moving bodies, which keep always 

 near one another (§ 23), passing near a projecting part of the fixed boundary, 

 and being, before and after this collision (§ 29), at a very great distance from 

 every part of the fixed boundary. Thus, a Helmholtz ring projected so as to pass 

 near a projecting angle of two walls, shows a deflection of its course, as if caused 

 by attraction towards the corner. 



36. In every case the force-resultant of the impulse is, as we shall presently 

 see (§ 37), determinate when the flow of the liquid across every element of any 

 surface completely enclosing the solids or vortices is given ; but not so, from such 

 data, either the axis (§ 6) or the rotational moment, as we see at once by con- 

 sidering the case of a solid sphere (which may afterwards be supposed liquefied) set 

 in motion by a force in any line not through the centre, and a couple in a plane 

 perpendicular to it. For this line will be the " axis," and the impulsive couple will 

 be the rotational moment of the whole motion of the solid and liquid. But the 

 liquid, on all sides, will move exactly as it would if the impulse were merely 

 an impulsive force of equal amount in a parallel line through the centre of the 

 sphere, with therefore this second line for " axis " and zero for rotational moment. 

 For illustration of rotational moment remaining latent in a liquid (with or with- 

 out solids) until made manifest by actions, tending to alter its axis, or showing 

 effects of centrifugal force due to it ; see § 66, and others later. 



37. The component impulse in any direction is equal to the corresponding 

 component momentum of the mass enclosed within the surface S, containing all 

 the places of application of the impulse, together with that of the impulsive 

 pressure outwards on this surface. But as the matter enclosed by S (whether all 

 liquid or partly liquid and partly solid) is of uniform density, its momentum will 

 be equal to its mass multiplied into the velocity of the centre of gravity of the 

 space within the surface S supposed to vary so as to enclose always the same 

 matter, and will therefore depend solely on the normal motion of S ; that is to 

 say, on the component of the fluid velocity in the direction of the normal at every 

 point of S. And the impulsive fluid pressure, corresponding to the generation of 

 the actual motion from rest, being the time integral of the pressure during the 

 instantaneous generation of the motion, is (§§ 31, 32) equal to — 0, the velocity 

 potential; which (§61) is determinate for every point of S, and of the exterior 

 space when the normal component of the fluid motion is given for every point of 

 S. Hence the proposition asserted in § 36. Denoting by da- an} 7 element of S ; 

 N the normal component of the fluid velocity ; a the inclination to OX, of the 

 normal drawn outwards through da- ; and X the ^-component of the impulse ; we 



