188 GEOPHYSICAL THEORY UNDER THE PLANETESIMAL HYPOTHESIS. 



As illustration may be considered the case of superficial strata deposited 

 when the earth had attained practically its final dimensions, and corre- 

 sponding therefore to the last entry in the tables. A stratum, for instance, 

 exerting the same pressure as a mercury column of 760 mm. would have a 

 thickness of 3.8 meters; the depression factor D is 0.737; hence that stra- 

 tum would depress the former surface 2.8 meters, giving actual increase to 

 the radius of only 1 meter. This increase bears to the total radius a ratio 



;r-j^XlO~*, while the value of , , o is 7.08; hence, for the case of normal 

 6.37 ^ ' d log /3 



impact, the corresponding increase of the period of rotation is about 

 -Q X 10"^ of the whole, or 0.096 of a second for a day of the present length. 



To increase the period just one second out of 86,164 would require a 

 stratum 39.7 meters thick. The illustration also indicates that under the 

 assumed law of compressibility the mean pressure of the atmosphere is 

 responsible for a diminution of the earth's radius amounting to 2.8 meters. 



The preceding computation of pressures and densities has been based 

 on the condition of hydrostatic equilibrium, which for any substance not 

 completely fluid can be considered strictly applicable only in case each por- 

 tion of the mass is subjected to compression in such a way as to avoid any 

 distortion of shape, which would call into play reactions against shearing 

 stresses. It will therefore be instructive to determine what kind of defor- 

 mation in the elementary portions of the mass is implied in the foregoing 

 account. In view of the radial symmetry assumed, the distortion at any 

 point in the completed planet may be expressed in terms of a distortion- 

 factor Sf defined as the ultimate ratio of vertical to horizontal dimensions 

 of a mass which when first deposited at the surface was cubical. 



The ratio of final to initial horizontal dimensions is /^J/?/, being the 

 same as the ratio of radii of two spheres passing through the same particles 

 at the respective epochs. The vertical or radial ratio in the same sense is 



^^wY> since conservation of the mass of a stratum implies 



if <?/?', d^ represent the respective thicknesses. These give for the distor- 

 tion-factor 



pMJ 



whose value at various depths is given in column 5 of table 4, which shows 

 that the vertical compression throughout exceeds the horizontal, the differ- 

 ence being most marked about one-fourth of the way to the center, where 

 the ratio is 0.824. This would be a violent deformation for a body with 

 perceptible rigidity, but may be admitted in the present theory if it be 

 supposed that under extremely slow changes the substance is practically 

 plastic, even if highly viscous. The chief uncertainty would then relate 

 to the energy of compression, which should take account of the work done 

 against viscosity under shear. Further comment on this matter will be 

 reserved for another place. 



