February 25, 1915] 



NATURE 



7^3 



If w were zero we should have a pendulum the 

 period of which is 2 7:*/llg for vibrations in the plane 

 of the paper supposed vertical, and 2 7r\'l'Jg for vibra- 

 tions at right angles to that plane ; and the motion 

 of the bob in the most general form would be com- 

 pounded of two such oscillatory motions. 



When the motion, with to not zero, is referred to 

 two rectangular horizontal axes of x and y through O, 

 which revolve with the rod, the equations of motion 

 of the bob are 



.r-2 ay + 



(5-') 



J + 2C0.V + f*^- «»2 W = o, 



where x, y are the co-ordinates of the bob and x, y 

 their time rates of change, and x, y are the accelera- 

 tions corresponding. 



The second terms on the left-hand sides of these 

 equations, —2ii)X, 2wy, are in form what were called 

 afterwards by Lord Kelvin gyrostatic terms, and the 

 conditions for the existence of real periods of oscilla- 

 tory motion in the general case, depending, as the 

 reality of these periods does, on the value of u>, gives 

 us an idea of what he termed in that connection 

 ■■gyrostatic domination." 



I'n an Appendix to this lecture (see Journal, I.E.E.) 

 will be found a synopsis of the solution of this interest- 

 ing case of motion with some modifications of nota- 

 tion and mode of presentment. The reader may refer 

 also to the original paper.'. It is reprinted as 

 Appendix F of Lord Kelvin's "Baltimore Lectures." 



The main results may be expressed as follows : — 



(i) If a long straight rod, which is unequally elastic 

 in two rectangular directions, or is of unequal dia- 

 meters in these directions, if of uniform elastic quality 

 (a rod of elliptic section, for example), be rotated 

 rapidly about its axis, and vibrations be maintained 

 in a fixed transverse direction at one end, waves of 

 rectilinear vibration, the direction of which slowly 

 turns round as the wave advances, will be propagated 

 along the rod. 



(2) Let the transverse elasticity of the medium 

 (which, to fix the ideas, may be taken, as has already 

 been suggested, as a long "straight rod, along which 

 waves of transverse displacement are propagated in 

 the direction of its length) var\- with the direction of 

 the transverse, so that it has maximum and minimum 

 values in axial planes at right angles to one another. 

 If this rod be slightly and uniformly twisted about its 

 axis, these planes become heli^oidal or screw surfaces. 

 Think now of a line in space parallel to the axis of 

 the rod. This line will intersect either of these surfaces 

 at points the successive distances apart of which are 

 all equal to the step 5 of the screw. If the rod be 

 , turned about its axis as a whole, each point of inter- 

 section will move along the line at a speed v which 

 depends on the rate of turning. 



Let a rectilinear vibration be kept applied at any 

 cross-section, say one end, and let the rod be rotated 

 about its axis in the proper direction, and at such a 

 rate that the speed v just specified is equal to the 

 velocitv of propagation of the wave produced by the 

 applied' vibration. The result will be that a series of 

 waves of rectilinear vibration will run along the rod, 

 without anv turning of the plane of vibration in space. 

 In order that the rotation may be rapid, it is neces- 

 sary that the wave-length, a say, should be many 

 times the step s of the screw. 



According to our notation the period of vibration is 

 2-/fx, and therefore the velocity of propagation of 



3 Zf-f. cil. above. 



NO. 2365, VOL. 94] 



the waves is a/i/2 7r. But if 5 be the step of the 

 screw, and u> denote as before the angular speed of 

 rotation, the value of i; is *>5 /2 r. Hence we must 

 have fi>s = a fi or ii) = a /x/s. 



The effects of the twist and rotation thus exactly 

 balance one another. The latter (see Appendix) gives 

 a rotation of amount ^ffA*/w*/i in a wave-length, or 

 a complete turn in 8/tw^/A* wave-lengths. Hence the 

 effect of a single turn of twist in a length s is 

 equivalent to that of rotation in 8/i«*/A* wave- 

 lengths. 



The dynamical illustration is thus applicable to all 

 the cases of turning of the plane of polarisation of 

 light. There is one point of difference, however, 

 which renders a rotational medium more truly repre- 

 sentative of the magneto-optic turning, and is decisive 

 as between a rotational and a structural explanation 

 of the different phenomena. A beam of plane polarised 

 light which has traversed a piece of heavy glass in a 

 magnetic field will, if it be reflected and sent back 

 through the medium, have the turning of the plane 

 doubled by the backward passage, while backward 

 passage through quartz or a sugar solution annuls 

 the turning produced by the forward passage. These 

 facts point, as Lord Kelvin repeatedly urged in his 

 teaching, to a rotational explanation of the magneto- 

 optic effect and to a structural explanation for the 

 other. 



III. — Precesstonal Motion of a Liquid. 



About twenty years later g^-rostatic problems at- 

 tracted Lord Kelvin's attention in a ver>- special way. 

 From 1875 onward for several years he was much 

 occupied with many things ; for instance, he transacted 

 much business connected with submarine cable instru- 

 ments, eclipsing lights for lighthouses, and compasses 

 and sounding machines. I was one of his assistants, 

 and remember how busy we all were. For the two 

 years from 1875 ^o 1877 there are set down in the 

 list of his papers fcur on the subject of gyrostatic 

 action, but of these only two Were ever printed, the 

 first and the last. The' former was entitled "Vibra- 

 tions and Waves in a Stretched Uniform Chain of 

 Symmetrical Gyrostats,"* the latter "On the Pre- 

 cessional Motion of a Liquid."' I shall first give 

 some account of the latter paper, because it contained 

 descriptions and illustrations of g>rostats and gyro- 

 static action, and shall then return to the former. 



The circumstances in which this paper was written 

 were interesting. In 1875 Lord Kelvin (then Sir 

 William Thomson) visited Ame-ica as one of the 

 judges of Group 25 (Scientific Instruments) of the 

 Centennial Exhibition at Philadelphia. He then met 

 and conferred on scientific questions with sonrie of 

 the most eminent natural philosophers of the United 



A conversation with Simon Newcomb, in Joseph 

 Henr)''s drawing-room in the Smithsonian Institu- 

 tion at Washington, led him to doubt the legitimacy 

 of some of his own conclusions regarding the effect 

 of elastic vielding of the crust on the precession and 

 nutation of a liquid earth contained within a solid 

 shell. These conclusions were stated in his paper on 

 the rigiditv of the earth,* and in §§ 847-8 of the first 

 edition of Thomson and Tait's " Natural Philosophy." 

 For example, he had decided that the yielding of the 

 crust of an internally liquid earth, under the differ- 

 ential attractions of the sun and moon, would pro- 

 duce an effect on the precession so great as to be 

 altogether incompatible with the excellent agreement 



* ProceeHifign of the London Mathenuitical Society, toL vi., p. 190. iS75» 

 Math, and Phys. Paper*. Tol. iv., p. 533. , o - 



5 British Association Report, 1876, Tnuisactions of Sections, p. t. 



6 Philosophical Transactions of the Royal Society, toI. chiu, p, 573, 1863- 



