1878.] changes on the Earth's axis of rotation. 163 



II. The axis of figure must continue sensibly coincident luith 

 the axis of rotation. 



Suppose the earth's shape to have so changed, that the axis of 

 rotation no longer coincides with that of figure. The instantaneous 

 axis of rotation, though as we have shown fixed in space, will no 

 longer be stationary in the body. It will be so moving that the 

 njioniental ellipsoid rolls on the tangent plane at its instantaneous 

 extremity. The path traced out by the instantaneous axis on 

 the surface of the momental ellipsoid (the polhode) will be an 

 oval with the end of the axis of figure in its centre. The momental 

 ellipsoid was originally a spheroid, and is therefore if slightly 

 deformed approximately or even actually still such. Hence the 

 oval will be approximately or actually a circle. Thus the in- 

 stantaneous axis may be considered to trace out on the momental 

 ellipsoid, and therefore also on the earth itself, a circle round 

 the pole of figure. We may notice that in space it is the axis of 

 figure which is moving ; the axis of rotation is fixed, as we showed 

 in Section I. 



Now if the pole of figure, owing to Geological changes, is 

 uniformly shifting its position in the body along a' straight line, 

 and if the pole of rotation is at any instant thus revolving 

 uniformly in a circle about it, the path (the roulette) it traces 

 out in the surface must be a trochoidal curve. The particular 

 shape is decided by the consideration that initially these two poles 

 were coincident. This shows that the trochoid is a cycloid. The 

 path then will be a series of cycloids, whose bases lie in the line 

 along which the axis of figure is being shifted. 



With the present shape of the earth, and the present value of 

 its moments of inertia, the period in which the pole of rotation 

 would describe its circle round the pole of figure would be about 

 SOO days. Small deformations of the earth can only produce 

 small changes in this period. Thus each cycloid would be com- 

 pleted in about 300 days, and the poles of figure and rotation 

 would coincide at intervals of the same length. Their greatest 

 separation would be at the middle of these intervals, and would 

 bear the same ratio to the distance moved over in that time by 

 the pole of figure, which the diameter of a cycloid does to its base, 

 i. e. 1 : TT. Internal changes such as we are at present acquainted 

 with, able to elevate or depress parts of the surface but a few feet 

 in a century, cannot in 300 days shift the pole of figure over a 

 serious distance. The widest separation between this pole and that 

 of rotation is we have shown less than one-third of this distance. 

 Thus the two poles, however they wander over the body, must 

 remain sensibly coincident. Evidently also our preliminary as- 

 sumption that the pole of figure should be moving in a straight 

 line becomes wholly unnecessary. 



Vol. III. Pt. iv. - 12 



