GEOLOGICAL CHANGES ON THE EAETH’S AXIS OF EOTATION. 
285 
equations, which may then be written 
da j, SL 
-dt~^= A’ 
, 8M 
X+^> = X' 
Now and c$M are the changes in L and M, when A-fa£, A+b£, C-\-ct are written 
for A, A, and C respectively. If £L, be thus formed, and the equations integrated, 
it will be found that the principal terms, arising from the sun’s attraction, are nine 
both in ^ and ^ sin 5 ; the same number of terms arise in the precession and nutation 
with respect to the plane of the lunar orbit, and these would have to be referred to the 
ecliptic. Sixteen out of the eighteen terms represent, however, only very small nutations, 
and the only terms of any interest are those which give rise to a secular change in the 
obliquity of the ecliptic. These terms may be picked out without reproducing the long 
calculation above referred to, for they arise entirely out of the constant couple acting 
about the equinoctial line, which gives rise to the uniform precession. 
Now this constant couple is CITm; whence L=CIIw sin M= — CTln cos nt. And 
Q 
since n involves — — - , therefore 
£L= — CII n (XX ^ s ^ n ^ — — CUn xx ^ cos n ^ 
If these be substituted in the equations of motion and the equations integrated, and 
only terms in sin nt in 6> 1} and those in cos nt in <y 2 , be retained, we get 
n b— c . II c— a 
•i- “ » C^A sm nt > •■= - » C= A cos nt 
Substituting in the geometrical equation — u l sin nt-\-a> 2 cos nt, and rejecting 
periodic terms, 
dd II a + b— 2c 
dt 2 n C— A 
8. General result with respect to the Obliquity of the Ecliptic. 
It was found in sec. 6 that the secular rate of change of 6, as due to the internal 
changes in the earth, was — ^. a + ^~ 2c . Since C— A is small compared to A, this term 
is small compared with the term found at the end of sec. 7. Hence, finally, taking all 
the terms together, we get the approximate result, 
dQ IT a + b— 2c 
dt = 2n % C-A * 
and for small changes in the obliquity, insufficient to materially affect H, 
„ • in 
0 — * + o ' 
a + b— 2c 
C-A 
