1876.] 



Geological Changes on the Earth's Axis. 



329 



lieves that it has not hitherto been treated at much length. The paper, 

 of which the following is an abstract, is an attempt to investigate the 

 results of the supposition that the earth is slowly changing its shape, 

 with especial reference to the effects on the obliquity of the ecliptic and 

 on the geographical position of the earth's axis of figure. 



1. This part of the paper is dem oted to the consideration of the pre- 

 cession and nutations of an ellipsoid of revolution which is slowly and 

 uniformly changing its shape. The change is supposed to proceed from 

 causes internal to the earth, and only to continue so long as the total 

 changes in the principal moments of inertia C and A remain small 

 compared to their difference, C— A. 



The problem is treated by means of M. Liouville's extension of Euler's 

 equations of motion of a rigid body about a point *. By an approximate 

 method these equations may be treated as linear, and the solution 

 divided into two parts. 



Let 6 be the obliquity of the ecliptic ; n cosec 6 the precession of the 

 equinoxes ; —n the angular velocity of rotation of the ellipsoid ; A-f at, 

 A-\-ht, the principal moments of inertia at the time t. Then it 



is shown that the secular effect on the obliquity of the ecliptic, as result- 

 ing from the motion of the principal axes in the body (which constitutes 

 the first part of the solution), is given by the equation 



dd ^ _n a + h-2G ^ 

 dt ~ A ' 



and as resulting from the change in the impressed forces, due to the 

 change of shape of the body (which constitutes the second part), is given 



by 



^ n a + 2c 

 dt~ 2n C-A 



The former part may be neglected compared with the latter. But 

 from such geological changes as we are entitled to assume in the case of 

 the earth, the total change in the obliquity of the ecliptic must be 

 exceedingly small. Even gigantic polar ice-caps during the Glacial 

 period could not have altered the position of the arctic circle by so much 

 as 3 inches ; and this is the most favourable redistribution of matter on 

 the earth's surface for producing that effect. Thus the obliquity of the 

 ecliptic has remained sensibly constant throughout geological history. 



It is also shown that, during any gradual deformation of the ellipsoid, 

 the instantaneous axis of rotation will always remain sensibly coincident 

 with the principal axis of figure. 



In the course of the work by which the previous results are attained 

 there is sho^Ti to be a small inequality in the motion of the instantaneous 

 axis, in consequence of which that axis describes a circle with uniform 



* LiouY. Journ. 2^ serie, t. iii. 1858, p. 1 ; Eouth's Eigid DjTiam. p. 150. 



