Angus/ 28, 1884] 



NA TURE 



419 



mathematical investigation of a gyrostatically dominated com- 

 bination contained in the passage of Thomson and Tait's "Natu- 

 ral Philosophy" referred to, it follows that any ideal system of mate- 

 rial particles, acting on one another mutually through massless con- 

 necting springs, may be perfectly imitated in a model consisting 

 of rigid links jointed together, and having rapidly rotating fly- 

 wheels pivoted on some or on all of the links. The imitation is 

 not confined to cases of equilibrium. It holds also for vibration 

 produced by disturbing the system infinitesimally from a position 

 of stable equilibrium and leaving it to itself. Thus we may 

 make a gyrostatic system such that it is in equilibrium under 

 the influence of certain positive forces applied to different 

 points of this system ; all the forces being precisely the same 

 as, and the points of application similarly situated to, those 

 of the stable system with springs. Then, provided proper 

 masses (that is to say, proper amounts and distributions 

 of inertia) be attributed to the links, we may remove 

 the external forces from each system, and the consequent vibra- 

 tion of the points of application of the forces will be identical. 

 Or we may act upon the systems of material points and springs 

 with any given forces for any given time, and leave it to itself, 

 and do the same thing for the gyrostatic system ; the consequent 

 motion will be the same in the two cases. If in the one case the 

 springs are made more and more stiff, and in the other case the 



angular velocities of the fly-wheels are made greater and greater, 

 the periods of the vibrational constituents of the motion will 

 become shorter and shorter, and the amplitudes smaller and 

 smaller, and the motions will approach more and more nearly 

 those of two perfectly rigid groups of material points, moving 

 through space and rotating according to the well-known mode 

 of rotation of a rigid body having unequal moments of inertia 

 about its three principal axes. In one case the ideal nearly rigid 

 connection between the particles is produced by massless exceed- 

 ingly stiff springs ; in the other case it is produced by the 

 exceedingly rapid rotation of the fly-wheels in a system which, 

 when the fly-wheels are deprived of their rotation, is perfectly 

 limp. 



The drawings (Figs. 1 and 2) before you illustrate two such 

 material systems. 1 The directions of rotation of the fly-wheels 

 in the gyrostatic system (Fig. 2) are indicated by directional 



' In Fig. 1 the two hooked rods seen projecting from the sphere are con- 

 nected by an elastic coach-spring. In Fig. 2 the hooked rods are connected 

 one to each of two opposite comers of a four-sided jointed frame, each mem- 

 ber of which carries a gyrostat so that the axis of rotation of the fly-wheel 

 is in the axis of the member of the frame which bears it. Each of the 

 hooked rods in Fig. 2 is connected to the framework through a swivel joint, 

 so that the whole gyrostatic framework may be rotated about the axis of the 

 hooked rods in order to annul the moment of momentum of the framework 

 about this axis due to rotation of the fly-wheels in the gyrostat. 



ellipses, which show in perspective the direction of rotation of 

 the fly-wheel of each gyrostat. The gyrostatic system (Fig. 2) 

 might have been constituted of two gyrostatic members, but 

 four are shown for symmetry. The inclosing circle represents in 

 each case in section an inclosing spherical shell to prevent the 

 interior from being seen. In the inside of one there are fly- 

 wheels, in the inside of the other a massless spring. The pro- 

 jecting hooked rods seem as if they are connected by a spring in 

 each case. If we hang any one of the systems up by the hook 

 on one of its projecting rods, and hang a weight to the hook of 

 the other projecting rod, the weight, when first put on, will 

 oscillate up and down, and will go on doing so for ever if the 

 system be absolutely unfrictional. If we check the vibration by 

 hand, the weight will hang down at rest, the pin drawn out to a 

 certain degree ; and the distance drawn out will be simply pro- 

 portional to the weight hung on, as in an ordinary spring 

 balance. 



Here, then, out of matter possessing rigidity, but absolutely 

 devoid of elasticity, we have made a perfect model of a spring 

 in the form of a spring balance. Connect millions of millions 

 of particles by pairs of rods such as these of this spring balance, 

 and we have a group of particles constituting an elastic solid ; 

 exactly fulfilling the mathematical ideal worked out by Navier, 

 Poisson, and Cauchy, and many other mathematicians who, fol- 



lowing their example, have endeavoured to found a theory of the 

 elasticity of solids on mutual attraction and repulsion between a 

 group of material particles. All that can possibly be done by 

 this theory, with its assumption of forces acting according to any 

 assumed law of relation to distance, is done by the gyrostatic 

 system. But the gyrostatic system does, besides, what the sys- 

 tem of naturally acting material particles cannot do : it consti- 

 tutes an elastic solid which can have the Faraday magneto-optic 

 rotation of the plane of polarisation of light ; supposing the 

 application of our solid to be a model of the luminiferous 

 ether for illustrating the undulatory theory of light. The 

 gyrostatic model spring balance is arranged to have zero 

 moment of momentum as a whole, and therefore to con- 

 tribute nothing to the Faraday rotation ; with this arrange- 

 ment the model illustrates the luminiferous ether in a field unaf- 

 fected by magnetic force. But now let there be a different rota- 

 tional velocity imparted to the jointed square round the axis of 

 the two projecting hooked rods, such as to give a resultant 

 moment of momentum round any given line through the centre 

 of inertia of the system, and let pairs of the hooked rods in the 

 model thus altered, which is no longer a model of a mere spring 

 balance, be applied as connections between millions of pairs of 

 particles as before ; with the lines of resultant moment of mo- 

 mentum all similarly directed. We now have a model elastic 



