98 Sir William Thomson on Vortex Statics. 



performs steady motion. Not so a body having three unequal 

 principal moments of inertia. 



(2) A rigid body of any shape, in an infinite homogeneous 

 liquid, rotating uniformly round any, always the same, fixed 

 line, and moving uniformly parallel to this line, is a case of 

 steady motion. 



(3) A perforated rigid body in an infinite liquid moving in 

 the manner of example (2), and having cyclic irrotational 

 motion of the liquid through its perforations, is a case of steady 

 motion. To this case belongs the irrotational motion of liquid 

 in the neighbourhood of any rotationally moving portion of 

 fluid of the same shape as the solid, provided the distribution 

 of the rotational motion is such that the shape of the portion 

 endowed with it remains unchanged. The object of the pre- 

 sent paper is to investigate general conditions for the fulfil- 

 ment of this proviso, and to investigate, further, the conditions 

 of stability of distribution of vortex motion satisfying the con- 

 dition of steadiness. 



3. General Synthetical Condition for Steadiness of Vortex 

 Motion. — The change of the fluid's molecular rotation at any 

 point fixed in space must be the same as if for the rotationally 

 moving portion of the fluid were substituted a solid, with the 

 amount and direction of axis of the fluid's actual molecular 

 rotation inscribed or marked at every point of it, and the 

 whole solid, carrying these inscriptions with it, were compelled 

 to move in some manner answering to the description of ex- 

 ample (2). If at any instant the distribution of molecular 

 rotation* through the fluid, and corresponding distribution of 

 fluid-velocity, are such as to fulfil this condition, it will be 

 fulfilled through all time. 



4. General Analytical Condition for Steadiness of Vortex 

 Motion. — If, with (§ 24, below) vorticity and "impulse" 

 given, the kinetic energy is a maximum or a minimum, it is 

 obvious that the motion is not only steady, but stable. If, 

 with same conditions, the energy is a maximum-minimum, the 

 motion is clearly steady, but it may be either unstable or stable. 



5. The simple circular Helmholtz ring is a case of stable 

 steady motion, with energy maximum-minimum for given 

 vorticity and given impulse. A circular vortex ring, with an 

 inner irrotational annular core, surrounded by a rotationally 

 moving annular shell (or endless tube), with irrotational cir- 



: * One of Helmholtz 's now well-known fundamental theorems shows 

 that, from the molecular rotation at every point of an infinite fluid, the velo- 

 city at every point is determinate, being expressed synthetically by the 

 same formulas as those for finding the " magnetic resultant force " of a pure 

 electromagnet. (Thomson's Reprint of Papers on Electrostatics and 

 Magnetism.) 



