Sept, 14, 1876] 



NATURE 



429 



and the lunar fortnightly nutation enormously, aflFected by in- 

 terior liquidity. 



But although so much could be foreseen readily enough, I 

 found it impossible to discover, without thoiough mathematical 

 investigation, what might be the characters and amounts of the 

 deviations from a rigid body's motion which the several cases of 

 precession and nutation contemplated would present. The 

 investij^ation, limited to the case of a homogeneous liquid 

 enclosed in an ellipsoidal shell, has brought out results which I 

 confess have greatly surprised me. When the interior ellipticity 

 of the shell is just too .<;mall, or the periodic speed of the dis- 

 turbance just too great to allow the motion of the whole to be 

 sensibly that of a rigid body, the deviation first sensible ren- 

 ders the precessional or nutational motion of the shell smaller 

 than if the whole were rigid, instead of greater, as I ex- 

 pected. The amount of this difference bears the same pro- 

 portion to the actual precession or nutation as the fraction 

 measuring the periodic speed of the disturbance (in terms 

 of the period of rotation as unity) bears to the fraction measur- 

 ing the interior ellipticity of the shell ; and it is remarkable that 

 this result is independent of the thickness of the shell, assumed 

 however to be small in proportion to the earth's radius. Thus 

 in the case of precession the effect of interior liquidity would be 

 to diminish the periodic speed of the precession in the propor- 

 tion stated ; in other words, it would add to the precessional 

 period a number of days equal to the multiple of the rotational 

 period equal to the number whose reciprocal measures the 

 ellipticity. Thus in the actual case of the earth if we still 

 take Ti\-f, as the ellipticity of the inner boundary of the supposed 

 rigid shell, the effect would be to augment by 300 days the pre- 

 cessional period of 2,600 years, or to diminish by about -^" the 

 annual precession of about 51" — an effect which I need not say 

 would be wholly insensible. But on the lunar nutation of i8"6 

 years period the effect of interior liquidity would be quite 

 sensible; l8 '6 years being 23 times 300 days, the effect would 

 be to diminish the axes of the ellipse which the earth's pole 

 describes in this period each by ^\ of its own amount. The 

 semi-axes of this ellipse calculated on the theory of perfect 

 rigidity from the very accurately known amount of precession 

 and the fairly accurate knowledge which we have of the 

 ratio of the lunar to the solar part of the precessional 

 motive are 9" '22 and 6" "86, with an uncertainty not amount- 

 ing to one-half per cent, on account of want of perfect 

 accuracy in the latter part of data. If the true values were less 

 each by ^^ of its own amount, the discrepance might have 

 escaped detection, or might not have escaped detection ; but 

 certainly could be found if looked for. So far nothing can be 

 considered as absolutely proved with reference to the interior 

 solidity of the earth from precession and nutation ; but now think 

 of the solar semi-annual and the lunar fortnightly nutations. The 

 period of each of these is less than 3CX) days. Now the hydro- 

 dynamical theory shows that irrespectively of the thickness of 

 the shell, the nutation of the crust would be zero if the period of 

 the nutational disturbance were 300 times the period of rotation 

 (the ellipticity being ^j) : if the nutational period were anything 

 between this and a certain smaller critical value depending on 

 the thickness of the crust, the nutation would be negative ; if 

 the period were equal to this second critical value, the nutation 

 would be infinite : and if the period were still less, the nutation 

 would be again positive. Farther the 183 days period of the aolar 

 nutation falls so little short of the critical 300 days, that the amount 

 of the nutation is not sensibly influenced by the thickness of the 

 crust — is negative and equal in absolute value to |-| (being the 

 reciprocal of \^ — i ) times what the amount woidd be were the 

 earth solid throughout. Now this amount as calculated in the 

 Nautical Almanac makes o"'55, and o"'Sl the semi-axes of the 

 ellipse traced by the earth's axis round its mean position ; and if 

 the true nutation placed the earth's axis on the opposite side of 

 an ellipse having "86 and "-81 for its semi-axes, the discrepance 

 could not possibly have escaped detection. But lastly, think of 

 the lunar fortnightly nutation. Its period is ^\ of 300 days, and 

 its amount, calculated in the Nautical Almanac on the theory of 

 complete solidity, is such that the greater semi-axis of the 

 approximately circular ellipse described by the pole is o" "0325. 

 Were the crust infinitely thin this nutation would be negative, 

 but its amount nineteen times that corresponding to solidity. 

 This would make the greater semi-axis of the approximately 

 circular ellipse described by the pole amount to 19 X o"*o885, 

 which is i"7. It would be negative and of some amount 

 between i"7 and infinity, if the thickness of the crust were any- 

 thing from zero to 120 kilometres. This conclusion is absolutely 



decisive against the geological hypothesis of a thin rigid shell full 

 of liquid. 



But interesting in a dynamical point of view as Hopkins's 

 problem is, it cannot afford a decisive argument against the 

 earth's interior liquidity. It assumes the crust to be perfectly 

 stiff and unyielding in its figure. This of course it cannot be, 

 because no material is infinitely rigid ; but, composed of rock 

 and possibly of continuous metal in the great depths, may the 

 crust not as a whole be stiff enough to practically fulfil the 

 condition of unj ieldingness ? No ; decidedly it could not : 

 on the contrary, were it of continuous steel and 500 kilo- 

 metres thick, it would jaeld very nearly as much as if it were 

 india-rubber, to the deforming influences of centrifugal force 

 and of the sun's and moon's attractions. Now, although 

 the full problem of precession and nutation, and what is now 

 necessarily included in it — tides, in a continuous revolving 

 liquid spheroid, whether homogeneous or heterogeneous, has 

 not yet been coherently worked out, I think I see far enough 

 towards a complete solution to say that precession and nutations 

 will be practically the same in it as in a solid globe, and that 

 the tides will be practically the same as those of the equilibrium 

 theory. From this it follows that precession and nutations of 

 the solid crust, with the practically perfect flexibility which it 

 would have even though it were 100 kilometres thick and as stiff 

 as steel, would be sensibly the same as if the whole earth from sur- 

 face to centre were solid and perfectly stiff. Hence precession and 

 nutations yield nothing to be said against such hypotheses as that 

 of Darwin, ' that the earth as a whole takes approximately the 

 figure due to gravity and centrifugal force, because of the fluidity 

 of the interior and the flexibility of the crust. But alas for this 

 ' ' attractive sensational idea that a molten interior to the globe un- 

 derlies a thin superficial crust ; its surface agitated by tidal waves 

 and flowing freely towards any issue that may here and there be 

 opened for its outward escape," as Poulett Scrope called it ! 

 the solid crust would yield so freely to the deforming influence 

 of sun and moon that it would simply carry the waters of the 

 ocean up and down with it, and there would be no sensible tidal 

 rise and fall of water relatively to land. 



The state of the case is shortly this : — The hypothesis of a 

 perfectly rigid crust containing liquid violates physics by assum- 

 ing preternaturally rigid matter and violates dynamical astronomy 

 in the solar semi-annual and lunar fortnightly nutations ; but 

 tidal theory has nothing to say against it. On the other hand the 

 tides decide against any crust flexible enough to perform the nuta- 

 tions correctly with a liquid interior, or as flexible as the crust 

 must be unless of preternaturally rigid matter. 



But now thrice to slay the slain ; suppose the earth this 

 moment to be a thin crust of rock or metal resting on liquid 

 matter. Its equilibrium would be unstable ! And what of the 

 upheavals and subsidences ? They would be strikingly analo- 

 gous to those of a ship which has been rammed : one portion of 

 crust up and another down, and then all down. I may say with 

 almost perfect certainty, that, whatever may be the relative 

 densities of rock, solid and melted, at or about the temperature 

 of liquefaction, it is, I think, quite certain that cold solid rock is 

 denser than hot melted rock : and no possible degree of rigidity 

 in the crust could prevent it from breaking in pieces and sinking 

 wholly below the liquid lava. Something like this may have 

 gone on and probably did go on for thousands of years after 

 solidification commenced ; surface portions of the melted mate- 

 rial losing heat, freezing, sinking immediately, or growing to 

 thicknesses of a few metres when the surface would be cool and 

 the whole solid dense enough to sink. "This process must go 

 on until the sunk portions of crust build up from the bottom a 

 sufficiently close- ribbed skeleton or frame, to allow fresh incrus- 

 tations to remain bridging across the now small areas of lava, 

 pools, or lakes. 



" In the honey-combed solid and liquid mass thus formed, 

 there must be a continual tendency for the liquid, in consequence 

 of its less specific gravity, to work its way up ; whether by 

 masses of solid falling from the roofs of vesicles or tunnels, and 

 causing earthquake shocks, or by the roof breaking quite through 

 when very thin, so as to cause two such hollows to unite or the 

 liquid of any of them to flow out freely over the outer surface of 

 the earth ; or by gradual subsidence of the solid owing to the 

 thermodynamic melting, which portions of it under intense stress 

 must experience accoiding to my brother's theory. The results 

 which must follow from ttiis tendency seem sufficiently great and 



' "Observations on the Parallel Roads of Glen Roy and other Parts of 

 Lochaber in Scotland, with an Attempt to prove that they are of Marine 

 Origin." — Transactions of the Royal Society for Feb. 1839, p. 8i, 



