GEOLOGICAL CHANGES ON THE EARTH’S AXIS OE ROTATION. 
277 
Let the angular velocity of rotation of the earth be —n, and let II cosec Q be the pre- 
cession of the equinoxes ; then in the small terms the following substitutions may be 
made : — 
<p =— (nt+e), cj y = — II cos(nt-\-e), u 2 =— Ilsin^-f-e), a 3 =—n, 
and the e may be omitted for brevity. 
Further, N (depending on the attractions of the sun and moon) is very small ; and a 
consideration of the third equation of motion shows that, when integrated, it leads to 
a 3 = — n- {- terms, which are very small during the limited period under consideration. 
And if these terms were substituted on the left-hand side of the two former equations, 
(J ^ 
they would be still further diluted by multiplication by the small quantities A 
and II. Hence the third equation may be neglected, and in the two former equations 
—n may be substituted for co 3 . 
C — A 
Making these substitutions, then, and writing p for A n, the equations become 
L n nf by) Tla . , , %) . . 
-^ja — b + c — -^j-tfsinwH-^- cos^+njy+^jsin^ 
_ (3 |(C ± o« 8 }_| (M+y); 
dw 2 M n n( ay) lib . / . %\ 
-dt+P® 1 = a “ X) ~ a + b + c — ->t cos sm^— H( y+-^ jcos nt 
f(C + c t)n-%) , 
+“| — a — 7 +x(”+A 
Then neglecting ~ compared to unity, putting C + ct=A in the small terms, and only 
retaining the more important terms, 
dco, L Tin , . Ila _ f fhq) • n 
-fa — (**> 2 =^— X < a ~ b-j-c}£ sinwi+-^- cos w£+IIjyi- -^jsm nt—n(3, 
^ ■ -f { — a + b + c j - 1 cos nt -f- ^ sin nt — II jy + ^ jcos nt + m *. 
These are the required equations of motion, and in integrating them they may be 
treated as linear. 
* If we wish to treat a, a, &c. as variable, we have only to add to the right-hand sides of these equations 
cos nt~-^ and 5 t sin nt ~ respectively. If we put L=1VI=0 and neglect f^ t , 
A Cot -ol Cot A Cot A Cot 
these equations will be found to be identical with the equations (2) given by Sir AY. Thomson in App. C. I 
had not noticed until it was pointed out by him, how nearly applicable my equations were to the case of 
varying velocities of distortion. — April 26, 1877. 
