484 ME. G. H. DAE WIN ON THE PEECESSION OE A VISCOUS SPHEROID, 
the tidal friction, and accordingly that it is equally uncertain at what rate the day is 
at present being lengthened; lastly, that if there is at present any change in the 
obliquity to the ecliptic, it must be very slowly decreasing. 
The result of this hypothesis of elastico-viscosity appears to me so curious that I 
shall proceed to show what might possibly have been the state of tilings a very long 
time ago, if the earth had been perfectly elastic for the tides of short period, but 
viscous for the fortnightly tide. 
There will now be no tidal friction, and the length of day remains constant. The 
equation of tidal reaction reduces to 
u°~ 3 „ 
/i—= — — -fh sin* i sm 4e 
0 
Here v? is a constant, being the value of — at the epoch; and u~- f-P is the value 
T 2 . ° 
of — at the time t. 
9”o 
The equation giving the rate of change of obliquity becomes 
thsnfHsmde 
Dividing the latter by the former, we have* 
And by integration 
sin idi=jjLd£ 
cos ?'=cos ifj — —l) 
If we look back long enough in time, we may find £=1 - 01, and /i being 4’007, we 
have 
cos f=cosf 0 —'04007 
Taking i 0 =23° 28', we find i=28° 40'. 
This result is independent of the degree of viscosity. When, however, we wish to 
find how long a period is requisite for this amount of change, some supposition as to 
viscosity is necessary. The time cannot be less than if sin 4e"= 1, or e" = 22° 30', and 
we may find a rough estimate of the time by writing the equation of tidal reaction 
d£ 3 v? . 
m 4 I, 
where I is constant and equal to 24°, suppose. Then integrating we have 
p,(P -1) = - sin 4 1, 
or 
t= -Uf, cosec* I(F-l). 
* Concerning the legitimacy of this change of variable, see the following section. 
