VIBRATORY MOVEMENTS AND THEIR EFFECTS. 47 
The explanation of rotations by means of crost vibrations seems first to have been given 
by F. Hoffmann’ and later repeated independently by Mallet ? and others, but it does not 
seem to have received the consideration it deserves. I think it is clear from this chapter 
that crost vibrations are not only capable of explaining rotations wherever the disturb- 
ance is sufficiently strong, but that no other theory, so far proposed, can explain satis- 
factorily the very large rotations which statues and monuments experience. 
SURFACE WAVES IN THE MEGASEISMIC DISTRICT. 
In addition to the ordinary vibrations which we have been studying, many persons 
reported waves in the ground which had the appearance of ordinary waves on the sur- 
face of water (vol. 1, pp. 380, 381). They were not a peculiarity of the California earth- 
quake, for similar phenomena have been recorded in connection with almost all great 
earthquakes and have given rise to much discussion as to their cause. It is probable that 
they result from the modifications of condensational vibrations by the surface, as appears 
from the following considerations. The resistance of a substance to compression and dis- 
tortion depends upon the values of two coefficients: k, the coefficient of compression under 
equal pressure in all directions, and n, the coefficient of rigidity or shear. If we compress 
a small cube of anysubstance between two plates, the modulus of compression, that is, the 
ratio of the applied forces per unit area to the linear contraction, is called Young’s 
modulus, and its value in terms of the coefficients mentioned above is er This 
represents the resistance which the substance offers to compression. When the pressure 
is exerted, the cube is not only comprest in the direction of the pressure, but it expands 
at right angles to this direction, and the ratio of this expansion to the normal compression 
. Sk-—2n 
 9k+n) 
it is not far from 1:4. When the vibrations pass thru the interior of the earth, the 
rocks are subjected to compressions and expansions, but the surrounding rock allows only 
longitudinal contraction or expansion and the modulus of elasticity is then given by 
Bk+4n 
3 
place upwards but not laterally, and it can be shown that here the modulus of elasticity 
4n(8k+n) 
3k+4n 
3k—2n 
3k+4n 
The values of k and n have been determined for a number of specimens of rock by 
Messrs. H. Nagaoka,* 8S. Kusakabe,* and Adams and Coker.’ The average values which 
the last investigators found for granites are k =4.3 x 10° pounds per square inch, and 
n =3 x 10° pounds per square inch; and the vertical expansion would be nearly 0.3 of 
the longitudinal compression. As we pass down from the surface the increasing weight 
of overlying rock would greatly diminish the vertical expansion, and at a depth compara- 
tively small would prevent it altogether. The actual vertical movement at the surface 
would be the addition of all the vertical expansions from the surface down. A longitu- 
dinal contraction of 1: 8,350, as found in the example already used, would cause a vertical 
The value of this ratio varies with different substances, but in general 
, which is greater than Young’s modulus. At the surface expansion can take 
is given by the expression , and the ratio of the vertical expansion to the 
longitudinal contraction is 

1 Nachgelassene Werke, 1838, vol. 11, p. 310. 
2 The great Neapolitan earthquake, vol. 1, pp. 375-381. 
3 Elastic Constants of Rocks and the Velocity of the Seismic Waves. Publications of the Earthquake 
Investigation Commission in Foreign Languages, No. 4, 1900; and Phil. Mag. July, 1900, vol. L. 
4On the Modulus of Rigidity of Rocks. Publications of the Earthquake Investigation Commission in 
Foreign Languages, No. 14, 1903; No. 17, 1904; and No. 228, 1906. 
5 An investigation into the Elastic Constants of Rocks, etc. Carnegie Institution of Washington, 
Publication No. 46, 1906. 
