On the Variation in the Length of the Day. 3:45 
fourth power of the radius in the above formula, does not express 
the true relation. - e es 
The angular velocity of a revolving body is represented by the 
— 
well known formula » = S(mr?)~ mk? 10 the sphere mp3} 
hence, since M and m’ in the same mass are identical, and since 
R and v and 2 are constants, © os and @ 3 o/; ‘pat pia There- 
fore, the angular velocities of a sphere of the same constant mass, 
but variable in volume, impelled by the same force, are inversely 
proportional to the squares of the radii, instead of the fourth 
powers, as given in the formula. 
Since the angular velocities are also inversely proportional to 
the times of rotation, the squares of the radii are proportional to 
2 aie 
the times of rotation, or r2 3 7/2114: ¢ and ell 'This is the 
formula that should have been used in the calculation on the 
638th page of the Geology of the Ist District of New York, as 
affording an approximation to the time of a revolution of the 
earth on its axis under the assumed condition of varying in di- 
ameter. 
If we apply this formula, supposing the radius of the earth to 
be one mile less than its present mean radius, the time of a revo- 
lution on its axis would be 234 5916”, or the day would be 
shortened about 44 seconds.* 
A diminution in the length of the day of one second would 
correspond to a diminished radius of about 40 yards.t| M. La 
Place has shown that the sidereal day, or true time of rotation of 
the earth, has not varied ;1, part of a centesimal second during 
2000 years. ‘To find what diminution of the mean radius cor- 
responds to this minute fraction of time, we have from the above 
formula 7’? = 
miles. h : 
r?t’ _(3956)? “ae = 315") Whence r—r’ is equal 
t 
fie an 
tr! 24(3955) 
25 | SO, ne a 2 pe 1 
a ee ee . 
t Prof. A. Ryors of the Ohio University, made this calculation about a year ago 
from the same formula here used, but deduced in a different way. Vide his lecture 
on Gravitation before the Chillicothe Lyceum.—Since this article was in type I 
have learned that the same formula is given in Poisson’s Mechanics, 2d edition, 
Tome II, p. 460. 
460 
Vol. xiv1, No. 2.—Jan.-March, 1844. 44 
