DUE TO THE WEIGHT OE CONTINENTS. 
203 
If this figure be made to rotate about the polar axis, the several curves will of 
course generate the surfaces of equal stress-difference throughout the sphere. 
Writing 3 for the inclination of one of the principal axes to the equator, we have 
by means of the formula (27) 
cot ^ = ~2T^ = ^{(r) — lj cosec 2$— cot 26 
It would be easy to trace out the changes of direction of the principal stress-axes 
throughout the sphere, but I will only now make the observation that ail over the 
surface they are parallel to and perpendicular to the surface, and that at the centre they 
are polar and equatorial, the stress-quadric being of course an ellipsoid of revolution. 
We have next to find the actual amount of stress-difference which arises from 
given ellipticity of form of the spheroid. Putting 7=2 in (30) we have 
< 5 i= sin 3 6—2 cos 3 9= 3[J— cos 3 d] 
The equation to the spheroid is 
r=a-\-hsi 
h 
1 ~ cos ~ Q) 
—— cos 3 oy ] 
Thus 3 hja is the ellipticity of the spheroid, which we may put equal to e. 
Then it was shown in (33) §4 that the results for the stresses P, Q, R, &c., are to 
be multiplied by —f givh/a?, and this will of course be also the factor for the stress- 
difference A. 
Then substituting e for 3 h/ct, and introducing the factor - x %, which has been omitted 
in considering the distribution of stress within the spheroid, we see that ellipticity c 
gives a stress-difference represented by 
4e 
A = —^P wa 
■t 
/ -] Z \ cos 26 
If we estimate the forces in gravitation units the factor g must be omitted. 
The expression under the square-root sign is equal to unity at the surface, and to 
8 at the centre. 
Thus the stress-differences, in gravitation units of force, at the surface and at the 
centre are - 9 - 5 -ewa and f-f ewa respectively. 
To apply this to the case of the earth, take a=637x 10 6 c.m., and w= 5*66, and we 
find the surface and central stress-differences to be respectively 152c and 1214c metric 
tonnes per square centimeter. 
If these numbers be multiplied by 6’34, we get the same result expressed in tons 
per square inch. Thus in British units these two stress-differences are 92 6e and 
7698c, 
