12 INTERNAL STRUCTURE OF THE EARTH 



and 13), viz., almost or quite insensibly minute nutations; but in this latter case, on 

 account of non-rigidity, they take the form of internal motion, or fluctuating distor- 

 tion, rather than of axial nutations. 



But while it is nidisputably true that the actual angular motion of " precession" 

 is at any instant just so much rotation-area converted into gyration-area, and while 

 the principle of areas is, in my opinion, a much surer basis on which to found an a 

 priori judgment as to the integrity of the precession of a fluid spheroid than that 

 of the quasi-rigidity conferred by vortex-motion ; yet even this basis may not seem 

 quite so solid for such a judgment, when we reflect that precession, in its universnl 

 acceptation (for a rigid body), as an angular movement always exactly normal to the 

 plane of a variable (in direction and intensity), tilting couple, has no relation hut of 

 antagonism to the principle of areas'; and that all our demonstrations of precession, 

 which commence with expounding the elementary gyration (e. (j.. Airy, "Figure of 

 the Earth," Encyc. Met.; Art, "Precession," Encyc. Britta., etc.), demonstrate, 

 apparently at least, something that is thus antagonistic: i. e., that an attractive force 

 produces motion directly normal to its own direction ; that it generates areas about 

 the line connecting its origin with the centre of inertia of the attracted body.* 



I have alluded to this anomaly in the note to page 44 of " Addendum," cited, 

 and the 5th and (ith pages of the first of the "Problems of Rotary Motion;" remark- 

 ing that ju'ecession can at no instant be exactly what it is computed to be, and that, 

 either in the whole as a rigid solid, or in the parts, as a yielding one, there must 

 be always going on " slight nutations" — the necessary concomitants of fluctuating 

 transfer of rotary into gyratory area (and the reverse), which must result from the 

 fluctuation in direction of plane and intensity of the tilnng couple. 



Incidentally it may be interesting to inquire how much these slight nutations 

 may amount to. For the earth, considered as perfectlt/ rigid, a nutximum limit can 

 be announced. Suppose at any instant the sun's declination be V, and the in- 

 stantaneous axis and axis of figure (or diurnal rotation) to be at rest; in other 

 words, tliat they coincide. Commencing with this initial state of things, the ear;h 

 will iirst yield to tlie sun-couple; rotation-ixrea. thus lost will transform itself into 

 gyration-area; and the path of the earth's axis of figure will be a continued cycloid 

 (see first of the "Problems of Rotary Motion," p. 3, and note; where u and are 



polar distances) of which the sagittce will be the excessively minute quantity — ^ 



' The denionstralions referred to in preceding paragrapli (said to originate witli Frisi), prove 

 really, and onh/, what must be tlie incipient motion, starting from rest, of the inslantaiieous axis 

 of the rotating splieroid subjected to a tilting action like that of the sun's attraction on the earth; 

 which incipient motion they mistake for what it is not (though in the case of the earth the actual 

 things differ inappreciably), its memi motion; and hence for the "Measure of the mean motion of 

 the Earth's axis." Starting with a conceded and dynamically calculated viotion of the axis oj 

 figure, they lose sight of t/hat axis entirely, and apparenlJy prove that this fundamental motion 

 never took place, but was really something el.'^e. The resulting expression for precession is made 

 formally wrong, by having as a factor the reciprocal of the moment of inertia about an equatorial 

 axis instead of about the axis of diurnal rotation ; and the above " measure'' would be quite inacca- 

 rate if applied to planets as oblate as Jupiter, Saturn, and Uranus (see paragraphs 3, 4, and 5, p. l; 

 those of fith page, and note 2 to p. 3 of "Addendum," "Problems of Rotary Motion"). 



