100 



Sir William Thomson on Vortex Statics. 



at three points 120° from one another, so as to make it into 

 as it were an equilateral triangle with rounded corners. Give 

 now a right-handed twist, round the radius to each corner, to 

 the plane of the curve at and near the corner ; and, keeping 

 the character of the twist thus given to the wire, bend it into 

 a certain determinate shape proper for the data of the vortex 

 motion. This is the shape of the vortex core in the second 

 steady mode of single and simple toroidal vortex motion with 

 rotational moment. The third is to be similarly arrived at, by 

 twisting the corners of a square having rounded corners ; the 

 fourth, by twisting the corners of a regular pentagon having 

 rounded corners ; the fifth, by twisting the corners of a hexa- 

 gon, and so on. 



In each of the annexed diagrams of toroidal helices a circle 

 is introduced to guide the judgment as to the relief above and 

 depression below the plane of the diagram which the curve 

 represented in each case must be imagined to have. The circle 

 may be imagined in each case to be the circular axis of a to- 

 roidal core on which the helix may be supposed to be wound. 



To avoid circumlocution, I have said "give a right-handed 

 twist" in each case. The result in each case, as in fig. 1, 

 illustrates a vortex motion for which the corresponding rigid 

 body describes left-handed helices, by all its particles, round 

 the central axis of the motion. If now, instead of right- 

 handed twists to the plane of the oval, or the corners of the 

 triangle, square, pentagon, &c, we give left-handed twists, as 

 in figs. 2, 3, 4, the result in each case will be a vortex motion 



Fig. 2. 



Fig. 3. 



Fig. 4. 



for which the corresponding rigid body describes right-handed 

 helices. It depends, of course, on the relation between the 

 directions of the force resultant and couple resultant of the 

 impulse, with no ambiguity in any case, whether the twists in 

 the forms, and in the lines of motion of the corresponding 

 rigid body, will be right-handed or left-handed. 



8. In each of these modes of motion the energy is a maxi- 

 mum-minimum for given force resultant and given couple 



