282 
ME. GEOEGE H. DAEWIN ON THE INELHENCE OP 
In consequence of the continuous deformation, the principal axis travels with a linear 
velocity (on the map) —y/ a 2 +0 2 along the meridian of longitude arc tan Take this 
meridian as axis of x, and measure y, so that the angular velocity p is from x towards y, 
and call ^/a 2 -}-^ 2 , u. 
Then the principal axis X 3 moves along the axis of x with a uniform linear velocity 
— u, and, from dynamical principles, the instantaneous axis I moves round the instanta- 
neous position of X 3 with a uniform angular velocity u>. 
But because of the earth’s viscosity, X 3 always tends to approach I. The stresses 
introduced in the earth by the want of coincidence of X 3 with I vary as X 3 I. Also the 
amount of flow of a viscous fluid, in a small interval of time, varies jointly as that interval 
and the stress. Hence the linear velocity (on the map), with which X 3 approaches I, 
varies as X 3 I (equal to r suppose). Let this velocity be vr, where v depends on the vis- 
cosity of the earth, diminishing as the viscosity increases. 
Thus the principal axis describes a sort of curve of pursuit on the map ; it is animated 
with a constant velocity — u parallel to x, and with a velocity vr towards I, which rotates 
round it with a uniform angular velocity p. 
The motion of I, relative to X 3 , is that of a point moving with a constant velocity u 
parallel to x , rotating round a fixed point with a constant angular velocity and moving 
towards that point with a velocity vr. 
Let £, 7i be the relative coordinates of I with respect to X 3 , and x, y the coordinates 
of X 3 . Then the differential equations which give the above motions are : — 
§=U~, f-p;, (1) 
§=-«+(*, ( 2 ) 
dx 
dt 
(3) 
dy 
~dt =r/l ' 
(4) 
If (1) and (2) be integrated, and the constants determined so that, when £=0, 0 
(which expresses that initially X 3 and I are coincident), it will be found that 
£=^-J 
\v(l — e vl cos [jjt)-\-pe vt sin 
u 1 
[jtfc(l —e~ vt cos [ht)—ve~ vl sin 
These give the path of I relative to X 3 . It may be seen to be a spiral curve dimi- 
nishing with more or less rapidity, according as the earth is less or more viscous. If 
y = 0, it becomes the circle, found above from the dynamical equations. 
