36 PRECESSION OF THE EQUINOXES AND NUTATION 



rigidity of the earth, may be several times as much as that of iron (which would 

 make the phenomena, both of the tides and precession, sensibly the same as if the 

 earth were perfectly rigid), it is enough that the actual rigidity should be several 

 times as great as the actual rigidity of iron throughout 2000 or more miles thick- 

 ness of crust." 



A theorem fundamental to the establishment of the above propositions is, that a 

 revolving spheroid destitute of rigidity, a homogeneous fluid one, for instance, 

 would have no precession. Sir W. Thomson docs not mathematically demonstrate 

 this theorem, but by use of an hypothesis gives an elegant illustration of its truth, 

 for which, though to me it is convincing, I prefer to substitute the following 

 demonstration. 



Such a spheroid, all the particles of which revolve about an axis with a common 

 angular velocity n, and attract each other by the law of universal gravitation, 

 would have the form of an ellipsoid of revolution, the ellipticity of its meridional 



c 2 



section being - ' .* (See " Figure of the Earth," Encyc. Metrop., par. 33, by Prof. 



Airy.) Attracted by the sun, its tides would be expressed by the terms of [2316] 

 Mec. Cel., Book IV (Bowditch). Of these three terms, the first (a function of the 

 declination wily of the attracting body) and the third (the semi-diurnal oscilla- 

 tion) express tidal elevations symmetrically distributed on each side of the equator, 

 which would, hence, exert no influence through the centrifugal forces of their 

 masses, upon precession. The second therefore, or the diurnal tide, is alone to be 

 considered. 



Conceive a meridian plane passed through the sun at any declination, the 

 " couple" exerted by its attraction would be exerted wholly to turn the spheroid 

 about an equatorial axis normal to this plane. We have therefore to investigate 

 what dynamic couple, with reference to this same axis, will be exerted by the cen- 

 trifugal force of the diurnal tidal protuberance. As the calculation involves the 

 state of things at but a single instant of time, the angle, nt-\-ss ^, may be written 

 is and counted from the meridian of the sun: p, the uniform density of the fluid, 

 taken as unity. The height, y, of the diurnal tide will be expressed for all parts 

 of the spheroid by 



(47) y = -_--,, sin0 cos0 sin^l cos a, cos of 



2r g 



in which % is the polar distance or complement of the latitude of the locality, and 

 the declination of the sun. If, with Laplace, we put cos X==, and sin X=l/l /u 2 , 

 the mass of the elementary column of height y will be ydfj. cZcr, and its centrifugal 



* g being the force of gravity at the equator of the hypothetical spheroid. 

 f The expression, in the original, for the diurnal oscillation, is 

 SL 

 s-; sin Fcos Fsin 9 cos 9 cos {nt4-v ^) 



The notation of my paper on the precession of the equinoxes is substituted, and the assumed value 

 of P introduced. 



