162 Mr E. Hill, On the influence of Geological [May 20, 



produced on the position of the Earth's axis by small deformations 

 in its shape. My object in these notes is to give comparatively 

 elementary discussions of some propositions which he has there 

 demonstrated. 



I. The Axis of Rotation is fixed in space. 



The angular momentum about an axis fixed in space can only 

 be changed by external force. From Newton's 3rd Law, or by 

 the Conservation of Angular Momenta, no internal actions can 

 produce any resultant momentum in a body. To change the axis 

 of rotation a new momentum must be produced to be compounded 

 with that previously existing. But no internal actions can produce 

 this. Thus no deformation produced by internal action can of 

 itself change the direction of the axis in space. 



And there is no external force to produce any considerable 

 alteration. The attractions of the sun or moon cannot do it. In 

 treatises on Precession and Nutation where the problem of their 

 effect on a rotating spheroid is worked out, it is shown that there 

 is no secular change in the obliquity. Hence any deformation 

 which leaves the earth a spheroid rotating about its axis of figure, 

 can only alter the magnitudes of periodic changes, and cannot pro- 

 duce any secular alteration of obliquity. We will show hereafter 

 that the deformations we deal with cannot make it rotate about 

 an axis different from that of figure. 



If the figure cease to be a spheroid, this appeal to the Pro- 

 cessional Theory must be modified. We may argue as follows : 



On a sphere's rotation the sun's attraction can produce no 

 effect, for it can exert no couple. On a spheroid rotating about 

 its axis of figure, the sun's attraction does exert a couple at right 

 angles to the plane of rotation, but none in this plane. The rate 

 of rotation therefore remains constant, and we know that the 

 above couple tending to decrease the obliquity, cannot per- 

 manently alter it. In an Ellipsoid this couple will vary periodi- 

 cally in magnitude, but this can only introduce a new periodic 

 term into the obliquity. A couple will however arise about the 

 axis of rotation, which may produce a permanent effect on the 

 rate, and, if so, possibly a secondary effect on the obliquity. Since 

 however the deviation from a spheroid is by supposition minute, 

 the change of rate of rotation must be very minute, and much 

 more so the above secondary effect on the obliquity, if indeed it 

 exist at all. 



This discussion is not very satisfactory, and it will be better to 

 use Mr Darwin's analysis. He also takes the spheroidal motion 

 as the first approximation, and his results also show that the 

 secular change in the obliquity only exists during the process 

 of deformation. 



