On the Physical G'eology of the United States, §c. 289 
the ocean in consequence of the inertia and increased centrifugal 
force, requiring an increased protuberance in the spheroidal form 
of the earth to restore the form of equilibrium to the revolving 
spheroid. 
The changes in the times of rotation of the earth being sup- 
posed paroxysmal, occurring at particular periods of time, and 
none of these having occurred during the historical epoch, (unless 
the time of the deluge was one of them,) the argument from the 
fact that the length of the day has not varied for two thousand 
years loses all its force, and cannot be adduced in opposition to 
the views here advocated. 
3d. It is well known that the angular velocities of a revolving 
spherical body under varied diameters, are inversely proportioned 
to the squares of the radii,* so that if the earth be a cooling body, 
* Vide M. Poisson’s Mecanique, second edition, Tome II, p. 460, and Ameri- 
can Journal of Science, Vol. xtv1, p. 344-346. 
The angular velocity ofa revolving body is represented by the well known formula 
__ MR» 
ae L(mr2) 
the moment of applied force divided by the moment of inertia. Y(mr2)=mk2, 
(Young’s Mechanics, Am. ed., p. 190,) in which k represents the radius of gyra- 
, (Young’s Mechanics, American edition, p. 193,) which is equal to 
¢ : avai MRv 
tion, and m the mass of the revolving body. By substitution w = ES In the re- 
volving sphere with a variable radius, the quantity of matter remaining constant 
i 2 1 
M=m, and as R and v are constants, #o-——. In the sphere k2 = =r2.. wo — 
k2 5 r2 
and 0! a Be acl: sii as or the angular velocities vary inversely as the squares 
520 : ye eae) gs yy, 
of the radii. 
If the variable density due to variable volume be considered, the law of the an- 
gular velocities being inversely as the squares of the radii stil] holds true; for, in 
4 2 
the sphere mk2 == amr? x ere when the density is unity. When the density is 
D, the moment of inertia is =D x zr xer2 and @ zoe HO Calling D’ the 
Lage ut 
density in the second place and r! the corresponding radius w! = be BRO es ; hence 
D/ —r!5 
X15 
o DI! o7l5 : xe : ‘ 
apa But the mass being constant, the densities are inversely proportioned 
7r!5 
ait val DV Ems , NN 
to their volumes, or Dow Substituting this value, we obtain ol yrs pp and 
by substituting for V and V! their values in terms of the radii, we obtain — 
@) 
gl5  l2 ok ee 
—_— SSS Ee oo: -. 
rd 72 r2 7!2 
Lieut. Roberts, Assistant Professor of Natural Philosophy, West Point, N. Y.) 
T3 
7/3 
(This last demonstration was communicated by 
