IN RELATION TO T1IK KAUTH'S INTERNAL STRUCTURE. 43 



tions : The fluid spheroid, treated of p. 36, is subjected, by the attraction of the 

 sun, to the distortion expressed by (47). This distortion, as shown by the form 

 of the expression, is equivalent to an exceedingly slight rotational displacement 1 

 "/ //////> about an equatorial axis, such as would be caused by displacing through a 

 still more minute angle the planes of diurnal rotation. It is one of the beautiful 

 results of the analy>is to show that the change in the direction of the centrifugal 

 force due to this slight obliquity of the planes of rotation is equivalent to turning 

 forces at all points of the fluid exactly proportional to their distances from the 

 equatorial axis. 



Let now a rigid shell, exactly conforming internally to the external surface of 

 the fluid, be applied, and the whole turned back until the planes of rotation are 

 restored to perpendicularity to their axis; the precessional effect of the attracting 

 body now takes effect upon the whole mass; for there is no longer a neutralizing 



centrifugal force. If we take that part of (47) which is due to the direct action 



o y 

 of the sun, viz., , sin 6 cos sin A, cos a cos or (for, the protuberances being repressed 



by the shell, the pressures on its interior which replace them will arise only from 

 the direct action), and estimate it as a pressure and calculate the elementary couples 

 for an internal ellipticity, e, we shall find the integral couple (and this corresponds 

 with Prof. Hopkins' result) to be identical with (48) viz., exactly that due to the 

 couple which the sun would exert on the fluid mass considered as a solid.* It would 



increase the precessional force of the shell in the ratio * of the analysis. By 



virtue of this pressure the fluid tends to transform its own precession into an 

 augmented precession of the shell. 



It requires, however, but an extremely minute angular separation of the axes of 

 the shell and fluid to generate counter-pressures equivalent to those which caused 

 the separation.* The divergence cannot, therefore, be progressive, but is simply a 

 minute oscillation of the two axes, or a rotation around each other. In the latter 



The required angle of this displacement is the height of the tidal wave (4T), for = 0, divided 

 by di~ f r " ellipse of elli P ticit 7> e, (2 e tnx cos x). Prof. Hopkins shows that such a divergence 



from perpendicularity develops a couple m, * n ' e multiplied by the sine of twice this arc (or 



24 S 

 twice the arc itself). Performing the operations we get, -*- t'n e cos e, in which we have the 



solar couple (19) and (48), which causes the displacement, since e g (C A). 



* The lever arm is also 2 e sin x cog x. Multiply the above by this arm, by g, by the elementary sur- 

 face, d n d , and, again, by cos , and we get the elementary component tending to tilt the shell. 

 The integral, with proper substitutions, is equivalent again to (19) or (48). 



1 There is another process which may take effect in neutralizing internal pressure. I have remarked 

 (last par. p. 6), that, considered as a perfectly rigid body, the precessional motions of the earth 

 cannot be precisely those assumed. In fact, our imperfect integrals of the conditional differential 

 equations present the anomaly of a varied motion in which the generating force does no work; no 

 yielding to the tilting couple having place. There are necessarily some, too minute to be detected, 



