SIR W. THOMSON ON VORTEX MOTION. 227 



solid or group of solids has been at an infinite distance from all other solids and 

 from the containing vessel — the time integrals of the impressed forces parallel 

 to three fixed axes, and of their moments round these lines, are equal to the six 

 corresponding components of" the impulse" (§ 6). 



27. If two groups, at first so far asunder as to exercise no sensible influence 

 on one another, come together, the "impulse" of the whole system remains un- 

 changed by any disturbance each may experience from the other, whether by im- 

 pacts of the solids, or through motion and pressure of the surrounding fluid; and 

 (§ 6) it is always reducible to the force-resultant along the central axis, and the 

 minimum couple-resultant, of the two impulses reckoned as if applied to one 

 rigid body. The same holds, of course, if one group separates into two so 

 distant as to no longer exert any sensible influence on one another. 



28. Hence whatever is lost of impulse perpendicular to a fixed plane, or of 

 component rotational movement round a fixed line, by one group through collision 

 with another, is gained by the other. 



29. Two of the moveable solids, or two groups, will be said to be in collision 

 when, having been so far asunder as not to disturb one another's motions sen- 

 sibly, they are so near as to do so. This disturbance will generally be supposed 

 to be through fluid pressure only, but impacts of solids on solids may take place 

 during a collision. 



30. We are now prepared to investigate (§§ 30, 31, 32) the influence of a fixed 

 solid on the impulse of a moveable solid, or of a vortex, or of a group of solids or 

 vortices, passing near it, thus — If during such collisions or separations as are con- 

 sidered in §§ 27, 28, forces are impressed on any one or more of the solids, their 

 alteration of the whole impulse is (§ 26) to be reckoned by adding to each of its 

 rectangular components the time integral of the corresponding component of 

 these impressed forces. Now, let us suppose such forces to be impressed on any 

 one of the moveable solids as shall keep it at rest. These forces are zero as long 

 as no moving solid is within a finite distance. But if a moving solid or vortex, 

 or group of solids or vortices, passes near the fixed solid, the change of pressure 

 due to the motion of the fluid will tend to move it, and the impression of force 

 on it becomes necessary to keep it fixed. Let da- be an element of its surface ; 

 (x, y, z), the co-ordinates of the centre of this element; a, /3, y the inclinations of 

 the normal at (x, y, z) to the three rectangular axes; and p the fluid pressure 

 at time t, and point (x, y, z). The six components of force and couple required 

 to hold the body fixed at time t, are 



If da . cos a . p , ffdt . cos |3 . p , ffde . cosy . p ; 7 



Jfdtr (y cosy — z cos j3)_p , ffde{z cos a — x cos y)p , ffde(x cos j6 — y cos x)p , j ^ '* 



If in these expressions we substitute 



fpdt (2). 



