On the Variation in the Length of the Day. 345 



fourth power of the radius in the above formula, does not express 

 the true relation. 



The angular velocity of a revolving body is represented by the 



MRv MRv MKv 



well known formula <» = ^7 —. = —-r^ =in the sphere — ,-— ; 



^(mr 2 ) m'k- r m\r 2 ' 



hence, since M and m' in the same mass are identical, and since 



111 



R and v and f are constants, w cc — and w : «': ;— ; — , 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 



tr' 2 

 the times of rotation, or r 2 \r f2 \\t\ V and t'=,—r-. This is the 



formula that should have been used in the calculation on the 

 638th page of the Geology of the 1st 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 23 h 59' 16", 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.f M. La 

 Place has shown that the sidereal day, or true time of rotation of 

 the earth, has not varied ji ? 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 



miles. h 



r 2 V (3956) 2 x (24-^1/0 

 formula r 12 =—— = ^ . Whence r—r' is equal 



h miles. 

 tr'2 24(3955)2 



= 23'' 59' 16" 



ra (3956)2 



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. 



Vol. xlvi, No. 2 Jan.-March, 1844. 44 



