GEOLOGICAL CHANGES ON THE EARTH’S AXIS OF ROTATION. 
293 
The problem is thus reduced to the following : — Rectangular axes are drawn at the 
centre of a sphere of radius c ; it is required to effect the above-described removal of 
matter, so that the product of inertia about a pair of planes through z, and inclined to 
xz at 45° on either side, shall be a maximum, subject to the above condition as to depth, 
k being small compared to c. For convenience, I refer to the plane xy as the equator, 
to xz as prime meridian, from which longitudes \J/ are measured from x towards y, and 
to Q the colatitude. These must not be confused with the terrestrial equator, longitude, 
and latitude. 
A little consideration shows that the seas and continents must be of uniform depth 
k, that there must be two of each, that they must all be of the same shape, must be 
symmetrical with respect to the equator, and that the continents must be symmetrical 
with respect to the prime meridian, and the seas with respect to meridians 90° 
and 270°. 
Also the total product of inertia P, produced by this distribution, is 16 times that 
produced by the part of one continent lying in the positive octant of space ; and the 
mass of matter removed is 8 times the mass of this same portion of one continent. 
The problem is, therefore, to find the outline of the continent, so that P may be a 
maximum, subject to the condition that the mass is given. 
Take the surface-density of the sphere as unity, and let the mass removed be given 
as an elevation of a height k over a fraction q of the whole sphere’s surface ; so that the 
mass removed from hollow to continent is 4 KC 2 kq. Then it may easily be shown that 
P = 4 kc x J 2 sin 3 $ sin 2 4 
and 
q=l J 2 2-4/ sin m, 
where \|/ is a function of $ to be determined. Then writing u for 2\|/, and (m for cos 3, 
we have to make 
fV-K 
i sm w — u cos" a 
}dyj a maximum, 
for it will be seen later that — c 2 cos 2 a is a proper form for the constant, to be introduced 
according to the principles of the Calculus of Variations. This leads at once to 
or 
(1 — qj 1 ) cos a = cos 2 a 
sin 2 $ cos cos 2 a, 
That is to say, the outline of the continent is the sphero-conic formed by the inter- 
section with the sphere of the cone, whose cartesian equation is 
y\l -\- cos 2 u)-\-z i cos 2 a=x 2 sin 2 a. 
Reverting to the expressions for P and q, altering the variable of integration, and the 
