820 
MR. G. H. DARWIN ON THE SECULAR CHANGES IN 
lo g e tan = log,, tan |J 0 + '04750 
log* tan ^1 = log e tan -|I 0 — '07953 
and on integration from 1 to '76 
log,, tan = log,, tan -|-J 0 + '02425 
log e tan ^1 = log,, tan -|I 0 — '23972 
Then if we take J 0 = 0°, I 0 =]7°, which are in round numbers the final values of J 
and I derived from the first method of integration, we easily find, 
when £=-88, J = 6° 17', 1=15° 43' 
and when £='76, J = 6° 9', 1 = 13° 25' 
Then we have by (249) 
• -j- “t • t b • T 
sin 1 = —— 4 - L sm J =-sm J 
a /c 2 + a. 
. T a . T /o+a . T 
sm J = -sm 4= ——-— sm 1 
' k 2 + a b 
Now b is always unity, and the logarithms of (k. 2 -\-u) and —(k, fi-a) are given in 
Table II.; from this we find 
when £='88, 1=1° 16', J=3° 39' 
when £='76, 1=2° 43', J=8° 54' 
By the same formula, when £=1 initially, we have ^ = 22', J =56'. These two 
results ought to be identical with the results from (260) of the last section ; and they 
are so veiy nearly, for at the end of the integration we had Si= 22' 43", SJ= 57' 31". 
The small discrepancy which exists is partly due to the assumed smallness of i and j in 
the present investigation, and also to our having taken the values 6° and 17° for J 0 
and I 0 instead of 6° 21', 17° 4'. 
The value £='88 gives the length of day as 8 hrs. 45 m., and the moon’s sidereal 
period as 5'57 m. s. days. 
The value £='76 gives the day as 7 hrs. 49 m., and the moon’s sidereal period as 
3‘59 m. s. days. This value of £ brings us to the point specified as the end of the 
third period of integration in § 17 of the paper on “Precession.” 
There is one other point which it will be interesting to determine,—it is to find the 
rate of the precessional motion of the node of the two proper planes on the ecliptic, 
and the rate of the motion of the nodes of the equator and orbit upon their respective 
proper planes. By means of the preceding numerical values, it will be easy to find 
these quantities at the epochs specified by £=1, '88, '76. 
