GEOLOGICAL CHANGES ON THE EARTH’S AXIS OE ROTATION. 
279 
them I found 1^=^ -, and 71^ = 0 ; and in the other ^ 1 =^ 2 =0. In order not to inter- 
rupt the thread of the argument, the calculation is given in Appendix A ; it will also 
be more intelligible after the latter part of this paper has been read. In the general 
case the same kind of proportion will subsist between fib and Aa, ftR and A/3, and we 
may therefore, without serious error, neglect the former compared with the latter. 
Thus, as far as concerns the present inequality, 
17 Ot 
GO*— — 
hB 
( 1 — cos pt) — I® sin fit. 
not 
sin (1 — cos (At). 
On account of this inequality the greatest angular distance (in radians) of the instan- 
taneous axis from the pole is — ' It will appear from the latter part of this 
paper that, if the elevation of a large continent proceeds at the rate of two feet in a 
century, \J a 1 + /3 2 may be about per annum, and [a is 360° in 306 days ; whence it 
follows that the greatest angle made by the instantaneous axis with the axis of figure is 
comparable with a quantity beyond the power of observation. On the score of 
these terms the instantaneous axis will therefore remain sensibly coincident with the 
axis of figure. 
They will, moreover, produce no secular alteration on the obliquity of the ecliptic, nor 
in the precession, because they will appear as periodic in ^ and ^ sin 0, with arguments 
n and n-V[b. 
Now although this inequality is so small, it nevertheless is of interest. 
If we map, on a tangent plane to the earth at its initial pole, the relative motion of 
the instantaneous axis and the pole of figure, we get, as the equation to the curve, 
— cos^)— ^ sin [At, 
y=pm ^+^(1 -cos [At). 
If t be eliminated from these equations, we get 
(*-:)'+( 
y+%) - 
2 + /3 2 
Thus the relative motion is a circle, passing through the origin, and touching a line 
inclined to the axis of y at an angle arc tan Therefore the instantaneous axis describes 
a circle passing through the pole of figure every 306th day ; and this circle touches the 
JVIDCCCLXXVII. 2 R 
