292 
MR. GEORGE H. DARWIN ON THE INFLUENCE OE 
limits, so as to exclude the imaginary parts of the integrals, we have as the equation to 
find a 
2=; J t ' cos x arc cos (Sv) d X’ 
and 
P=4 fee* 1 cos % \/ cos 4 ^— cos 4 a d%. 
J o 
These integrals are reducible to elliptic functions ; but in order not to interrupt the 
argument, I give the reduction in Appendix B. If cos 2y=cos 2 «, the result is that 
^=^ C ^[n'(-2 S inV)-F'] 
or 
2= l-l {E'F-F(F-E)} 
and 
P 8^2 
^ 4 = -3- COS y [E 1 - cos 2yF*], 
where the modulus of the complete functions E 1 , F 1 , IT 1 is tan y, and where E, F have 
a modulus and an amplitude 5 — y. 
It will be observed that a is the semi-length of the continent in latitude, and y the 
semi-breadth in longitude. 
From these expressions I have constructed the following Table : — 
Semi-breadth of 
continent 
(r). 
Semi-length of 
continent 
(•). 
Fraction of surface 
elevated or depressed 
(S'). 
Product of inertia 
& 
0 
O i 
0 
•0000 
•0000 
5 
7 5 
•0054 
•0672 
10 
14 13 
•0216 
•2628 
15 
21 28 
•0486 
•5697 
20 
28 55 
•0867 
•9603 
25 
36 42 
•1362 
1-3981 
30 
45 0 
•1979 
1-8399 
35 
54 12 
•2732 
2-2371 
40 
65 22 
45 
90 0 
•5000 
2-6667 
15. Application of preceding problem to the case of the Earth. 
In the application to the case of the earth, what has been called, for brevity, the 
equator (EE in fig. 2) must be taken as a great circle, passing through a point in 
terrestrial latitude 45°. 
Figure 2 gives the stereographic projection of the forms of continents and seas 
the firm lines showing continents, and the broken ones seas, when covering various 
