476 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
present diurnal rotation, and when the moon is moving in her present orbit, and no 
distinction will be made between n and n 0 ; all the secular changes will be considered 
later. 
The numerical data of Section 11 are here used, and the obliquity of the ecliptic 
t — 26° 28'; then u and u / being expressed in radians per tropical year, I find 
n 
c' 
2'7561 
10 5 
E sin 2 < 
•6143™ . ,1 
-E sin e 
10 5 
M 14978 . -2669 . , 
G,= i ^rsm2 £ + — £ S me 
!• • 
■ (44) 
Tlien integrating the equation (37) and putting n=n 0 , when t — 0 
n. 
n =Bo -y=nol. -q-i 
• ( 45 ) 
Integrating a second tune, we find that a fixed meridian in the earth has fallen 
behind the place it would have had, if the rotation had not been retarded, by 
;fi )£ 2 648000 , „ . , , in 
E ^ - • — -— seconds ol arc. And at the end ot a century it is behind time 
1900*27i£ sin 2e+423'49 E' sin e'm. s. seconds of time. 
If the earth were purely viscous, and when e=17° 30'* (which by Section 11 
causes the rate of change of obliquity to be a maximum), I find that at the end of a 
century the earth is behind time in its rotation by 17 minutes 5 seconds. 
By substitution from the second of (44), equation (45) may be written in the form 
/, 14978 v . -2669 . , 
n — n Q [ 1— tt sm 2 e- sm e 
. . . (46) 
which in the supposed case of pure viscosity when e=l7° 
30' becomes 
006460 \ 
io« 7 
(47) 
All these results would, however, cease to be even approximately true after a few 
millions of years. 
The effectvof the failure of the earth to keep true time is to cause an apparent 
acceleration of the moon’s motion; and if the moon’s motion were really unaffected by 
* This calculation was done before I perceived that I had not chosen that degree of viscosity which 
makes the tidal friction a maximum, but as all the other numerical calculations have been worked out for 
this degree of viscosity I adhere to it here also. 
