GEOLOGICAL CHANGES ON THE EAETH’S AXIS OE ROTATION. 
283 
Substitute in (3) and (4) for <£ and t;; integrate, and determine the constants, so that 
when £=0, x=y=0. It will then be found that 
*= ~7Tp + (M+ Tpp | - (?-p)e- vt cos pt+ 2^e- vt sin pt\] , 
y=jrpp [— pve~ vt cos pt+(v 2 -p)e~ vt sin pi }] . 
These give the path of X 3 on the map. It may be seen to be a cycloidal curve, in 
which the radius of the rolling circle diminishes with more or less rapidity, according 
as the earth is less or more viscous. 
After some time e~ vt becomes very small, and the motion is steady ; and then 
g=^— 5 , 7 j = or I is fixed, relatively to X 3 , at a distance 7 =—- Q from it, and on 
p-\-v i v- + /x“ Vv z -\-p 
the meridian, measured from the axis of x, in longitude arc tan This point is the 
centre of the above-mentioned spiral curve. 
If v be very small (or the earth nearly rigid) this meridian differs by little from the 
axis of y. But it may be that v is so small that e~ vt has not time to become insensible 
before the geological changes cease. This case corresponds very nearly to the hypo- 
thesis, in the last section, of adjusting earthquakes. 
If the earth be very mobile, or v large, §=-, 7j=0. 
Again, with respect to the path of X 3 , when the motion has become steady, 
«v(v 2 — p) pu 
X P + p V 2 + fJt - 2 ’ 
2 jj.uv ' 2 [xuv 
y v 2 -f- p v 2 -{■ p 9 
and eliminating t , vx-] -pj— — uP. 
That is to say, when the motion is steady, X 3 moves parallel to meridian longitude 
t— arc tan and distant from it ~ n on the negative side. This straight line is the 
P Vv* + P & ° 
degraded form of the above-mentioned cycloidal curve. 
If the earth is nearly rigid this path does not differ sensibly from the axis of x ; if 
very mobile, it is nearly perpendicular to the axis of x, and a long w r ay from the origin. 
In this last case the solution becomes nugatory, except as showing that the very small 
inequality of 306 days would be capable of disturbing and quite altering the path of the 
principal axis, as arising merely from geological changes on the surface of the earth. 
In the case contemplated by the Astronomer Royal, where the elevation is explosive, 
u must be put equal to zero, and the constants of integration so determined, that when 
t= 0, £=R suppose, and q=x=y= 0. It will then be found that when the agitation has 
