14 



NATURE 



[May 2, 1878 



burgh, that the product of their mutual distances must be 

 a maximum or minimum or a maximum-minimum for a 

 given value of the sum of the squares of their distances 

 from the common centre of gravity;^ and the condition 



4 • # 



imslaile 



stahle 



Fig. «. 



Fig. 3. 



that this kinetic" equilibrium may be stable is that the 

 product be a true minimum for a given value of the sum 

 of the squares of their distances from the centre of 

 inertia. Taking for example a triad of vortices (or of 

 the little magnetic needles of Mr. Mayer's problem), it 

 is thus obvious the equilibrium is unstable in the case 

 represented by Fig. 2, and stable in the case represented 

 by Fig. 3. The arrow-heads in Figs. 2 and 3 represent 

 the motions of the vortex columns round their centre of 

 gravity. It must be understood that the core of each 

 column revolves also round its centre of gravity in the 

 same direction as the group round the common centre of 

 gravity of all with enormously greater angular velocities. 



I have farther considered the problem of oscillations in 

 the neighbourhood of configuration of stable equilibrium. 

 The general problem which it represents for mathemati- 

 cal analysis has a very easy and simple solution for the 

 case of a triad of equal vortex columns in the neighbour- 

 hood of the angles of an equilateral triangle. 



A mechanism for producing it kinematically is repre- 

 sented in Fig, 4, showing three circular discs of cardboard 

 pivotted on pins through their centres at the angles of an 

 equilateral triangle rotating in a vertical plane. The 

 plane carrying these three centres may be conveniently 

 made of a circular disc of stiff cardboard, or of hght 

 wood pivotted on a fixed pin through its centre. Each 

 of the small discs or epicycles is prevented from rotation 



Fig. 4- 



by a fine thread bearing a weight, and attached to a point 

 of its circumference ; and on each of them is marked, by 

 a small dark shaded circle, the section of one of the 

 vortex cores in proper position. 



' Helmholtz proved that whatever be the complication of motions due to 

 mutual influences among the vortices, their centre of gravity must remain 

 at rest. 



The rule for placing the vortices on their epicycles is 

 as follows : — Each vortex keeps a'constant distance from 

 its mean position (this being the centre of the epicycle, 

 carrying it in the mechanism) ; each of the radius vectors 

 drawn from the centres of the epicycles to the centres of 

 the vortices keeps an absolutely fixed direction, while the 

 equilateral triangle of the centres of the epicycles rotates 

 uniformly ; and these three fixed directions are inclined 

 to one another at equal angles of 120° measured back- 

 wards relatively to the order in which we take the three 

 vortices. It is easily verified that when the distances of 

 the vortices from their mean positions are infinitely 

 small (that is to say, when the triangle of the triad 

 is infinitely nearly equilateral), the product of its three 

 sides remains constant in the movement actually gfiven 

 by the mechanism, and so does the sum of the squares 

 of the distances of its three corners from its centre of 

 gravity. From the stability of the equilateral triangle it 

 follows' that there must be stability with three equal vor- 

 tices at the corners of an equilateral triangle, and one 



(whether equal to them or not) at its centre,* . For 

 four equal vortices I have found that the square order, 

 , also is stable. From the stability of the square fol- 

 lows (for vortices or for particles repelling according to 

 inverse distance) the stability of four equals at the corners 

 of the square, and one (whether equal to them or not) at 



its centre,^ • I have not yet ascertained waM^wrt/^V^//)' 



whether for^a pentad of equal vortices there is stability also 



in the pentagonal arrangement, * ' But Mr. Mayer's 



experiment, showing it to be stable for the magnets, is 

 an experimental proof that it must be stable for the vor- 

 tices, for it is easily proved that if any of the figures is 

 stable with mutual repulsion varying more rapidly (as is 

 the case with the magnets in Mr. Mayer' s experiment), 

 than according to the inverse distance, a fortiori, it must 

 be stable when the force varies inversely as the distance. 

 From the stability of the pentagon I infer (for vortices 

 and for particles repelling according to inverse distance) 



the stability of the configuration " . 



Mr. Mayer' s figure * . . shows that the hexagonal 



order was unstable for his six magnets. I had almost con- 

 vinced myself before seeing the account of his experi- 

 ments in Nature, that the hexagonal order is stable for 

 six equal vortices; and Mr. Mayer's last figure shows 

 that with his magnets the hexagonal order is rendered 



stable by the addition of one in the centre . . . 



The instability of the hexagon of six magnets shows 

 the simple polygon to be unstable for seven or any other 

 number exceeding six. Thus Mr. Mayer's beautiful 

 experiment brings us very near an experimental solution 

 of a problem which has for years been before me un- 

 solved — of vital importance in the theory of vortex atoms 

 — to find the greatest number of bars which a vortex 

 mouse-mill can have. 



William Thomson 



1 in the case of vortices or of the static problem when the law of the 

 mutual repulsions is the inverse distance, but not with the law of repulsion 

 with ordinary proportions of linear dimensions and magnetic distributions, in 

 Mr. Mayer's magnetic arrangement. 



2 In repetitions of Mr. Mayer's experiments, I have always found this 

 configuration unstable, and for four only the square stable. 



3 This configuration of the floating magnets I have found stable, but with 

 less wide limits of stability than the pentagon. 



* I have not found this, nor any other configuration than the pentagon 

 with centre, stable for six floating magnets. 



