GEOLOGICAL CHANGES ON THE EARTH’S AXIS OF ROTATION. 
289 
Now this expression must be identical with the former for all values of l, m, n ; hence 
putting 1=1, m=n = 0, A^A-j-a, and similarly B'=B-l-b, C'=C+c. Wherefore also 
and 
D D 
a — — C-B 
nearly, 
^ A-C ’ 7 B 
F 
-A ’ 
and these are the required expressions for a, (3, y. 
If, however, B= A, y becomes infinite, and the solution is nugatory : but since, under 
this condition, all axes in the plane of xy were originally principal axes, the axes of 
reference may always be so chosen that F is zero absolutely ; and then 
a= C— A’ 0= 
C-A 
, y=0. 
Therefore the new principal axis C' is inclined to the old C at a small angle , 
and is displaced along the meridian, whose longitude, measured from the plane of xz, 
is 7r+arctan^. This is the case to be dealt with in the present problem. The posi- 
tions of the other principal axes will be of no interest. 
12. Application of preceding Theorem. 
To solve the problem numerically in any particular case, it will be necessary to find 
the integrals 
D=h%c 4 JJ F(0, <p) sin 2 0 cos 0 sin <p dQ d<p, 
E=7igc 4 jj F(0, <p) sin 2 0 cos 0 cos <p dQ dip. 
D E 
If 4 and ^4 be called d and e, then d and e stand for the above integrals, which 
depend on the distribution of surface-matter in continents and seas. 
It will be convenient to use a foot as the unit for measuring h, and seconds of arc for 
the measurement of the inclination i of the new principal axis to the old. For this 
purpose the value of the coefficient ma y be calculated once for all. Let its value 
when multiplied by the appropriate factors for the use of the above units be called K *. 
Now 
C— A=f Kl+f,)(«-“ )mc 2 . 
* I have to thank Prof. J. C. Adams for his help with respect to the numerical data, and for having dis- 
cussed several other points with me. 
2 s 2 
