AS AFFECTING PRECESSION ASD NUTATION. H 



periods, as thoy are thus detcrminod, arc not independent of tlie thickness of the 

 shell; though for the semi-monthly (lunar) nutation they would correspond, about, 

 to the conventional 40 or 50 miles of the "thin crusl" of geologists, and hence should 

 produce sensible effeots in connection with that otherwise insensible nutation. Hence 

 the last paragraph of the "Addendum" to paper cited, and one of its summary "con- 

 clusions," affirms that "noticeable nutational movements" would be the concomitants 

 of a " thin and rigid shell;" for which even the ordinary small oscillations might be 

 distinguishable, and from which by the probable approximation of period to that of 

 semi-monthly nutation, extraordinary magnitudes might ensue. That "conclusion," 

 therefore (tlie 5th), expressed a test, even thougli not declared, " absolutely decisive 

 against the geological hypothesis of a thin rigid shell full of liquid;" while it had 

 no such test-application to a tJuck one for which the ordinary magnitudes of oscilla- 

 tion would be nndistinguishably minute, while extraordinary ones due to coales- 

 cing in period with the semi-annual solar nutation require particular values of shell- 

 thickness restricted to narrow limits. 



1 have already stated that a revision of my published investigations was under- 

 taken in consequence of suggestions made by Prof. Newcomb, last August. He 

 suggested tliat the law of "conservation of areas" would be found to carry with it 

 an identical precession for solid and liquid spheroid. Quite familiar with the law 

 in this connection, I was nevertheless prompted to inquire how far the internal 

 motion of the fluid in forming the tidal protuberances would affect the " areas." I 

 had already discovered (as they have been now described in preceding pages) what 

 these motions must be (p. 43 and note "Problems," etc.); but I have to thank 

 Prof. Newcomb for directing my tlioughts toward this matter again. I soon per- 

 ceived that these internal motions, bi/ changing the direction of the repelling force, 

 completely upset my demonstration of nil precession ; and this without direct refer- 

 ence to the "conservation of areas." 



To appreciate now the application of tliat principle ; conceive the sun in the 

 equinox — when v (the declination) is zero — the deviation of the planes of rotation 

 are then zero, and the precessional motion is assumed to he zero. No areas are now 

 generated save those of the earth's diurnal rotation, Cn; the component of which 

 about the line of the sun's attraction is zero. Now as the sun's right ascension and 

 declination increase, tliis component of the diurnal-rotation-area develops itself, 

 pari passu, in magnitude. But a (with declination acquired) tilting force would be 

 also exerted by the sun ; the earth, if perfectly rigid, will yield to it, to the impair- 

 ing of this component, causing an area-comp>cnsating movement, gyrafion (which is 

 elementary precession), to ensue. Suppose now instead of the body's yielding as a 

 perfectly rigid body, the jj»7anes qf rotation individually tilt. I have shown, else- 

 where ("The Gyroscope analytically examhied"), that while the initial motion of 

 gyration is an actual tilting, this latter effect is quickly checked by its own offspring, 

 gyratimi. So it will be also with reference to tlie tilting of these planes, which may or 

 may not reach the limit of angular displacement denoted by the minute angle (2ty) 

 required to develop the diurnal tide of the equilibrium theory. Tlie principle of 

 " areas" would, therefore, involve for the fluid s])heroid, what it does in a rigid 

 earth (as the fact is set forth in tlie first of tlie "Problems of Rotary Motion," pp. 6 



