136 SOME APPLICATIONS OF 
of the same isotropic elastic structure throughout, rotating with uni- 
form angular velocity » about its polar axis. 
Let a denote the mean radius, d the difference between the equatorial 
and polar semiaxes of the surface, EH Young’s modulus, and 77 Poisson’s 
ratio for the material. Then the ratio d: ais given for various values 
of 77 in the following table :* 
TABLE I. 
| ly hagas Rielle | 
| n 0 0.2 0.25 0.3 04 | 0.5 
| 
| ag | | | a vi = aml | 
| 2 | | | | } 
Pee OO maa | 0.330 | 0.341 | 0.352 | 0.373 | 0.395 
a E | | | | | 
i 
In the case of an originally spherical solid assuming the shape of the 
earth under rotation, it is of no practical importance whether we re- 
gard «as the radius of the original spherical surface, or as the mean 
radius under rotation, nor does it matter practically whether the density 
be supposed uniform previous to or during the rotation. There is, it 
is true, for all values of 7 except 0.5, a slight increase in the volume,t 
and consequent diminution in the mean density accompanying the ro- 
tation, but for our present purpose this may be neglected. 
The mathematical solution on which Table I is based treats the 
spherical surface of radius a as that over which the conditions for a 
free surface are satisfied. Now,some uncertainty may exist, depending 
on the physical interpretation put upon the mathematical equations, 
whether these surface conditions should be applied over what is the 
surface before the displacement—in this case the surface of the true 
sphere which it is assumed the earth would form if the rotation dis- 
appeared—or over what is the surface during the rotation. This un- 
certainty might constitute a very serious difficulty if the deformations 
were supposed to be large, a contingency which may arise when the 
limitation (C) in the magnitude of the strains is neglected; but in sueh 
problems as the present where the strains are, as we shall see presently, 
of the same order of magnitude as occur in ordinary engineering strue- 
tures, it is of no material consequence. In the present case complete 
assurance on this point may be derived from Figs. 1 and 2, Pl. 1m of 
(¢c), which show the changes induced by rotation in the equatorial and 
polar semi-axes of spheroids of various shapes. 
For given values of d, a, w, and p, Table 1 shows that E and 7 in- 
crease together. Giving @ the value it has for the earth, and assuming 
p=d).9, W=3900, d=13.25, I find for the values of HK, measured in grams 
weight per square centimeter, answering respectively to the values 0, 
0.25, and 0.5 of 7, the approximate numbers 
1020 x 10°, 1220x108, and 1410x108 
“See (a) formula (5) p. 287; or (c) Tables m1, v, and vi. 
t See (b) Table 11, and compare Tables v and vi of (c). 
