AS AFFECTING PRECESSION AND NUTATION. 7 



mise would make the fluid— not a " quasi" but — an absolute rl(jid body, in which 

 case wc know that the rotation would introduce, in connection with this tiltin"- the 

 usual phenomena of gyration, and hence of unmodiiied precession. But a fluid, 

 even under influence of vortex-motion, is not aaluaUy a rigid body, and hence an 

 answer must be sought for in studying the internal motions by which, under the 

 influence of rotation, the protuberance is developed. These have already been 

 indicated for the secular and for the semi-diurnal tide. Let us turn now to the 

 diurnal. 



It is a result of Mr. Hopkins' investigation that, given a fluid rotating vithin a 

 shell, internally an oblate ellipsoid of revolution of small cllipticity, e, with uniform 

 angular velocity, n, about the axis of figure ; if the axis of the shell be tilted. Fig. 2, 

 from coincidence with that of the fluid through any very small angle, y, the induced 

 derangement of the fluid will be a tiltimj of the planes of rotation through a still 

 slighter angle, viz., through tlie very minute angle 2ey. (A reference to an ellipse 

 of small ellipticity, e, will illustrate the converse fact that a slight and equal tilting 

 (say 2ey) of the longer diameter and of all parallels to it, about their middle point 



will produce a magnified tilting of figure equal to ^^= y.) Now, as I have 



e 



already shown, the diurnal tidal development is precisely tliat which such a tilting 

 (through y, putting expression {l)) = y) of the external surface (^regarding that surface 

 as a rigid hut infiniteJg thin shell) would cause. Hence the internal motion of tlio 

 fluid mass by which this diurnal development takes place is discovered to be tliis 

 slight change of direction, ■2ey, of the jjlanes of j-otatiou* — the centres a, a, a, etc., 

 of each rotating circular area still remaining steadfast in the fixed axis of the fluid 

 mass. Mr. Hopkins further shows that the resistance to this distortion is a couple 

 measured by 2ey into y^jTt «'.* We have found y for the diurnal tide to be (see 



expression (b) ), , > siny cos v. The resisting couple is therefore, = ^^ 7te_^siny 



* Mr. Hopkins' conclusion bas reference exclusively to the shell-enveloped fluid; the angle y not 

 only having no reference to foreign attraction but being entirely arl^itrary. I applied it inune- 

 diately (note p. 2 of Addendum to paper already cited) to the diurnal tidal development of the earth 

 considered as wholly fluid. Mr. Hopkins enters into a supplemental investigation (rhil. Trans., 

 1839, § 33) to ascertain the "degree of approximation," finding that the poxaible disturbance cannot 

 differ materially from that described above. It is moreover the least possible derangement by which 

 the requisite change of figure can be produced ; and, in fact, tlu; fluid cannot move otherwise and 

 preserve its vis viva of rotation. How " slight" the change of direction required for these planes 

 will be appreciated, when it is said that the angle y (sec expression h) is but U seconds of arc for the 

 sun ; and 3 seconds for the moon. 2ey, the change of direction of the planes is therefore ji^ of these 

 small arcs. The pressure couple on the shell due to the angular displacement y is found l)y Mr. 

 Hopkins to be -j^fteabi^ sin 2y; which may be written ^%na^yen-. Since, a (mean radius of earth) is 

 unity, and y very minute, the above expression results. The above may be written An-ye (A being 

 the moment of inertia). The last three factors are self evidently such, since the couple would vanish 

 with either; and the moment of inertia is obviously another. Mr. Hopkins finds, too, that for homo- 

 gcneousness this is also the couple exerted on the confined rotating fluid by the divei-ted "centrifugal 

 force." It is, however, but the " reaction" on the fluid of the shell against the above pressure couple 

 exerted on it by the fluid. His failure to recognize an identity which for homogeneonsness is ex- 

 hibited by his own analysis, is the clue to the fallacy of his most important results. (See Appendix.) 



