6 REPORT 1876. 



(4) Lastly, that the lunar semiannual nutation must he largely, and the lunar 

 fortnightly nutation enormously affected by interior liquidity. 



But although so much could be foreseen readily enough, I found it impossible 

 to discover without thorough mathematical investigation what might be the cha- 

 racters and amounts of the deviations from a rigid body's motion which the several 

 cases of precession and nutation contemplated would present. The investigation, 

 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 small, or the periodic speed of the distiu bance j ust 

 too great to allow the motion of the whole to be sensibly that of a rigid body, the 

 deviation first sensible renders the processional or nutational motion of the shell 

 smaller thau if the whole were rigid, instead of greater, as I expected. The amount 

 of this difference bears the same proportion to the actual precession or nutation as 

 the fraction measining the periodic -speed of the disturbance (in terms of the period 

 of rotation as unity) bears to the fraction measuring 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 eflect of interior liquidity would be to diminish the periodic 

 speed of the precession in the proportion stated ; 'in other words, it would add to 

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

 period equal to the number whose reciprocal measures the ellipticity. Thus, in the 

 actual case of the earth, if we still take — as the ellipticity of the inner boundary 

 of the supposed rigid shell, the effect would be to augment by 800 days the pre- 

 cessional period of 2G00 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 hmar nutation of 18'6 years period, the effect of interior liquidity would be 

 quite sensible; 18-6 years being twenty- three times 300 days, the eflect would be 

 to diminish the axes of the ellipse which the earth's pole desciibes in this period 



each by gg of its own amount. The semiaxes 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 luotion, are 9"-22 and 6"'86, -n-ith an uncertainty not 

 amounting 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 dis- 

 crepance 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 abso- 

 lutely proved with reference to the interior solidity of the earth from precession 

 and nutation ; but now think of the solar semiannual and the lunar fortnightly 

 nutations. The period of each of these is less than -'jOO days. Now the hydro- 

 dynamical theory shows that, in-espectively of the thickness of the shell, the nuta- 

 tion of the crust would be zero if the period of the nutational disturbance were 



300 times the period of rotation (the ellipticity being .■^) ; if the nutational period 

 were any thing 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. Further, the 183 days period of 

 the solar nutation falls so little short of the critical SCO 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 js^— 1) times what 

 the amount would be were the earth solid throughout. Now this amount, as 

 calculated in the 'Nautical Almanac,' makes 0"-55 and 0"'51 the semiaxes 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 0"-86 and 0"-81 for 

 its .semiaxes, 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 thee ry of complete solidity, is 



