10 INTERNAL STRUCTURE OF THE EARTH. 



precession of shell and fluid is, therefore, necessarily attended with minute relative 

 oscillations which manifest themselves as a slight periodical "inequality" in the 

 rate of precession of the shell, and also more noticeably as an equally slight 

 periodical " nutation" of its axis. They are, even for a thin shell, and, a fortion, 

 for a thick one (excepting always the special cases alluded to in what follows), 

 rerf/ minute, "not rising to magnitudes greater than those of the order of solar 

 nutation"^ {i. e., about one-half second of arc). 



The period of these oscillations, as generalized from Mr. Hopkins' expressions, 

 is equal to that fraction of the number of days denoted by the reciprocal of the 

 internal ellipticity of the shell, which expresses the ratio of moment of inertia of 

 shell to that of the whole mass. The period, therefore, for a (jiren internal ellip- 

 ticity, varies directly with the moment of inertia of the shell, which, itself, is greater 

 or less as its thickness is greater or less. 



A critical value for this period, discussed by Mr. Hopkins, is one which differs 

 but slightly from that of solar (semi-annual) nutation (implying a crust thickness 

 of something over 1000 miles); in which case there is, as might be expected from 

 the synchronism, a secular inequality in the solar nutation which may reach great 

 magnitude. A corresponding critical value may happen for the (to observation 

 inappreciable) lunar semi-monthly nutation (which does not enter into Mr. Hopkins' 

 results, and is not discussed by him) ; but there can be (as I understand the matter) 

 no critical value of this kind connected with the c7/(>/and sole recognized astro- 

 nomical nutation (the lunar nineteen yearly), unless the interior ellipticity be mwc/t 

 less than the gi^ (that of the earth's surface), and practically nil.'^ These critical 



' The astronomical nutations with which comparison is made are totally distinct in their origin 

 from these theoretical nutations arising from reciprocal oscillations of shell and fluid, of which the 

 rationale has just been given; as they are, also, from the "slight nutations" mentioned in subse- 

 quent pages. They (the astronomical) are but subsidiary manifestations of the general gyratory 

 motion caused by the sun and moon in their varying relations of declination to the earth's equator. 



^ Unless the internal ellipticity be such as to constrain the liquid and shell to the same mean 

 precessional motions the phenomenon of "precession" passes beyond the range of our power ot 

 prediction. The separation of the rotation axes (supposed initially coincident) of shell and fluid 

 becomes greater and greater. Friction or viscosity come into powerful action with results we cannot 

 define, except indeed to predict a gradual wasting of " vortex-motion" of fluid, and loss of rotatiou 

 of solid shell. With diminution of internal ellipticity the independent precessional co-efficieut of the 

 shell becomes greater than that of the whole mass; the "deviations first sensible" when the ellipticity 

 becomes too small to constrain to unison of precession, would appear to me to be those of acceleration. 



The case of an infinitely thin crust (of nil moment of inertia, of course) is identically that of the 

 wholly fluid spheroid discussed in the text; the deductions to be made referring not to internal motion 

 but io figure-moWon of the surface (see p. Y). The "infinitely thin rigid shell" moves through the 



whole angle 7 ; that is, it moves through „— times the angle to which the fluid planes of rotation 



must tilt to develop the diurnal tide. The angle y we have found to be about 1^ and 3 seconds for, 

 respectively, the sun and moon. Such a motion, if fully developed (see p. 13 and note), could 

 only obtain as a fluctuation of short (daily) period ; but reason has been given (ibid.) why this angle 

 cannot be fully developed. If the theorem of this present paper is correct, precession and the astro- 

 nomical nutations (which are but parts of precession) would be wholly unafi"ected. In the foregoing 

 deductions concerning the secondary phenomena of (rigid) shell precession and nutation, I am not 

 in entire unison with Sir Wm. Thomson, in his Glasgow Address. 



