490 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
Also, multiplying tlie equation of obliquity ( 66 ) by —, we have 
lo gio 6 
sm ^ cos i 
Lu^, n e di 1 dt V(id , A /t-. . . , ^ ^ 
ia-ism*,-) 7rm Sr +U ') ( sm +Q8111 } 
~~(U sin 4e+Y sin 2e) — ^ It sin 4e" 
dp 
Now by far the most important term in is that in which W occurs, and therefore 
1 dp 
— only depends on the obliquity in its smaller term. Then, since 2W=cos 2 i, 
therefore 
Also 
— = —A ( 2W— 
d £ cos 2 i\ d% 
cos- % 
. , di—d . lot 
sm % 
sin i cos i (1 — f sin 2 i) ' y/1 —f sin 2 i 
= d . loge tan i (1 — ^ sin 3 i) 
when the fourth power of sin i is neglected. 
Hence the equation may be written 
lo 8 ’iotan i( 1 —1-sin 3 i) = ~[2W^ uf j(P sin 4e+Q sin 2 c) 
— t^Rsin 4e // —~(U sin 4e-fV sin 2e')J.(75) 
dt\V / u 
Now the term in P (which is a constant) is by far the most important of those 
within brackets [ ] on the right-hand side, and 2 W^ has been shown only to involve 
i in its smaller term. Hence the whole of the right-hand side only involves the 
obliquity to a subordinate degree, and, in as far as it does so, an average value may 
be assigned to i without producing much error. 
In the equation of tidal reaction ( 68 ) or (74) also, I attribute to i in W and X an 
average value, and treat them as constants. As the accumulation of the error of 
time from period to period is unimportant, this method of approximation will give quite 
good enough results. 
We are now in a position to track the changes in the obliquity, the day, and the 
month, and to find the time occupied by the changes by the method of quadratures. 
First estimate an average value of i and compute Q, It . . . Z, /3, y. Take seven 
values of viz. : 1, '98, '96 . . . ' 88 , and calculate seven corresponding values of N\ 
then calculate -.seven corresponding values of sin 4e, sin 2 e', sin 4e". Substitute these 
values in —, and reciprocate so as to get seven equidistant values of . 
