AS AFFECTING PRECESSION AND NUTATION. 13 



COS v; or (referring to value of /:?, when it is interpreted for tlie case of the earth's 



(7 ^ 



precession in the note, putting — j^^e.) 



, . Gils' 



(r) — — , e sin v cos v. 



r ir , 



This, by what seems a singular coincidence, is identically the product of our 

 angle y (see expression (/>)) by 2e' ; in other words, it is exactly the angle %'y to 

 which our planes of rotation of a fluid spheroid must tilt to form tlie diurnal tide; 

 the precession affecting part of the tidal protuberance. 



This very minute angle (at its greatest about Jg second of arc for the moon, 

 lialf that for the sun) is tlie maximum to which the rUjid earth could be actually 

 tilted; but the axis of figure is never (except at the equinoxes) in this supposed 

 initial state of rest. It (see " Problems," etc., p. 6) always has approxiinately the 

 gyratory motion {i. e , parallel to the chord nearly of the cycloid) in which the tilting 

 effect is suppressed, viz., that of the mean gyration due to tlie sun's relative posi- 

 tion. The small nutations will be then but infinitesimal fractions of the minute 

 angle 2ey, or (c), which we have found to be that of the tilting of the planes of 

 rotation necessary to the production of the diurnal tide. 



When we turn now to i\\e fluid spheroid we have found that the distortion just 

 mentioned, and which constitutes the diurnal tide, is necessary to the full action 

 of tlie sun-couple as upon a solid and per/ectlj/ rigid spheroid. It would seem, there- 

 fore, that the diurnal tide (as are the other forms of tide) would be, as Sir Wm. 

 Thomson conjectures, fully developed — /. c, "practically the s;inie as that of the 

 equilibriuni theory."' On the other hand, this tilting of the fluid planes of rota- 

 tion involves precisely the same transfer uj' rotation into iji/ration area as the equal 

 tilting of the rigid spheroid ; and this is double^ the amount due to the mean gyra- 

 tion corresponding to the sun's declination ; for when (see " Problems," etc., where 

 above cited) the axis of figure is at the lowest point of the cycloid its gyratory 

 velocity is double the mean value. 



The precessional motion thus generated is a fluctuation ; it waxes from zero 

 to double its mean value and again wanes; the axis of figure making synchronously 



' This dictum, liowever, seems hardly a legitimate revelation of the light east l)y the "tlieory of 

 quasi-rigidity ;" for the tidal distortion of the equilibrium theory being all that follows perfect fluidity, 

 it renders " quasi-rigidity" a " nonieu inane." 



' This may be directly demonstrated : The normal component of rotation-area per unit of time 

 about the solstitial line, is Cm sin v. By the tilling of the planes this would be reduced to Cn sin 



fv e sin v cos v), i. e., a decremenl of area, (since e ^ — ^ — ), of-^ (£7 — ^)sin vcos'v. 



This requires a compensating gyration (i. e., motion about the solstitial line of ^ — '— sin v, 



which is double the mean gyration (elementary precession) of the earth (sec 20, p. 1, "Problems," 

 etc.). It must be observed that were sueh a tilting of the planes momentarily to have place (i. e., 

 that the diurnal tide be fully developed), it could only be as a phase of a periodic Jiucluation 

 corresponding to the cycloidal movement of the rigid spheroid (" Problems," etc., pp. 3 and 13), the 

 "period" being about one day, dilTering, however, in character from the diurnal tide:. 



