512 
ME. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID 
The change in this quantity from one epoch to another is the amount of moment 
of momentum of the moon-earth system which has been destroyed by solar tidal 
friction. This destroyed moment of momentum reappears in the form of moment of 
momentum of the moon and earth in their orbital motion round the sun. 
Now at the beginning of the integration of Section 17, that is to say at the present 
time, I find that when the present year is taken as the unit of time, the resultant 
moment of momentum of the moon and earth is 11369 C. 
At the end of the third period of integration (after which the solar terms were 
neglected), and when the obliquity has become 15° 22', I find the same quantity to be 
1 1625 C. 
Hence the loss of moment of momentum is 256 C., or 102'4 Ma~. 
Now at the present time the moment of momentum of the moon and earth in their 
orbit is (M-\- ni)/2cf—Mcft. ; - is clearly the sun’s parallax, and with the 
present unit of time /2, is 27 t. 
Hence the loss of moment of momentum is equal to the present moment of 
_ , (sun’s parallax) 3 . 
Z7T 1+r - L ' 
102'4 v 
momentum of orbital motion multiplied by — 
But the moment of momentum of the earth’s and moon’s orbital motion round the 
sun varies as /2 / '; hence the loss of moment of momentum corresponding to a change 
S/2 
of Z2 to Z 2 / J r 8 f 2 / is the present moment of momentum multiplied by xNT, whence it 
J2/ 
is clear that 
8/2 102’4 v , , ,, ,, 
—— = 3—— • -— • X (sun s parallax)" 1 . 
Ji / iir 1+v 
But the shortening of the year is —p of a year; taking therefore the sun’s parallax 
as 8"*8, we find that at the end of the third period of integration the year was shorter 
than at present by 
102-4 82 / 8-8tt 
3X 2tt X 83 X 1648,000 
X 365’25 X 86,400 seconds, 
which will be found equal to 2 ’77 seconds. 
Thus the solar tidal reaction had only the effect of lengthening the year by 2f seconds, 
since the epoch specified as the end of the third period of integration. The whole 
change in the length of year since the initial condition to which we traced back the 
moon would probably be very small indeed, but it is impossible to make this assertion 
positively, because, as observed above, the solar effects must have again become sensible, 
after passing through a period of insensibility. 
