580 
MR. G. H. DARWIN ON PROBLEMS CONNECTED 
Wi lo s tan " i 
1 - + ( ~) 2 - (^') + K [iX(l - + X(2 - X)(^) 2 + i( 1 - 3 x) (3—2X) (^f)“ 
dN 
44 ) 
i-^(;') 3 +i|©+KM2-4-') 2 
( 71 ) 
If K be put equal to zero, we have the equations (84) which were the subject of 
integration in Section 17 “Precession.” 
Since K, X, and t~ -f- r~ are all small, the correction to the second equation is 
obviously insignificant, and we may take the term in K in the numerator of the first 
equation as being equal to j;K(l — 2X)(3 — 2X)(t / -Pt). This correction is small although 
not insensible. This shows that the amount of change of obliquity has been slightly 
under-estimated. It does not, however, seem worth while to compute the corrected 
value for the change of obliquity in the integrations of the preceding paper. 
The equation of conservation of moment of momentum, which is derived from the 
integration of the second of (71), clearly remains sensibly unaffected. 
We see also from (70) that the time required for the changes has been over¬ 
estimated. If Iv 0 , X 0 ; K, X be the initial and final values of K and X at the beginning 
and end of one of the periods of integration; then it is obvious that our estimate of 
time should have been multiplied by some fraction lying between 1 — K 0 (l—X 0 )~ and 
l-K(l-X) 2 . 
Now at the beginning of the first period K 0 =‘0364 and X 0 —’0365, and at the end 
K=‘0865 and X= - 0346. 
Whence K 0 (l —X 0 ) 2 =‘034, K(1 — X) 2 = '080. 
Hence it follows that the time, in the first period of the integration of Section 15, 
may have been over-estimated by some percentage less than some number lying- 
bet ween 3 and 8. 
In fact, I have corrected the first period of that integration by a rather more tedious 
process than that here exhibited, and I found that the time was over-estimated by a 
little less than 3 per cent. And it was found that we ought to subtract from the 
46,300,000 years comprised within the first period about 1,300,000 years. I also found 
that the error in the final value of the obliquity could hardly amount to more than 1' or 2'. 
In the later periods of integration the error in the time would no doubt be a little 
larger fraction of the time comprised within each period, but as it is not interesting to 
find the time in anything but round numbers, it is not worth while to find the 
corrections. 
There is another point worth noticing. It might be suspected that when we 
