LAPLACE. 167 



Will then vary in the opposite direction, and the obliquity will continually 

 increase in like manner as it previously diminished. Finally, the in. 

 clination of the equator and the ecliptic will attain a certain maximum 

 value, and then the obliquity will again diminish. Thus the angle con- 

 tained between the two planes will perpetually oscillate within certain 

 limits. The extent of variation is inconsiderable. Laplace found that, 

 in consequence of the spheroidal figure of the earth, it is even less than 

 it would otherwise have been. This will be readily understood, when 

 we state that the disturbing action of the sun and moon upon the ter- 

 restrial spheroid produces an oscillation of the earth's axis which occa- 

 sions a periodic variation of the obliquity of the ecliptic. Now, as the 

 plane of the ecliptic approaches the equator, the mean disturbing action 

 of the sun and moon upon the redundant matter accumulated around 

 the latter will undergo a corresponding variation, and hence will arise 

 an inconceivably slow movement of the plane of the equator, which will 

 necessarily affect the obliquity of the ecliptic. Laplace found that if it 

 were not for this cause, the obliquity of the ecliptic would oscillate to 

 the extent of 49 53' 33" on each side of a mean value, but that when the 

 movements of both planes are taken into acccunt, the extent of oscilla- 

 tion is reduced to 1° 33' 45". 



Variation of the length of the tropical year. — The disturbing action 

 of the sun and moon upon the terrestrial spheroid occasions a continual 

 regression of the equinoctial points, and hence arises the distinction 

 between the sidereal and tropical year. The effect is modified in a small 

 degree by the variation of the plane of the ecliptic, which tends to produce 

 a progression of the equinoxes. If the movement of the equinoctial points 

 arising from these combined causes was uniform, the length of the tropi- 

 cal year would be manifestly invariable. Theory, however, indicates 

 that for ages past the rate of regression has been slowly increasing, and 

 consequently the length of the tropical year has been gradually dimin- 

 ishing. The rate of diminution is exceedingly small. Laplace found 

 tliat it amounts to somewhat less than half a second in a century. Con- 

 sequently the length of the tropical year is now about ten seconds less 

 than it was in the time of Hipparchus. 



Limits of variation of the tropical year modified by the disiurbing 

 action of the sun and moon upon the terrestrial spheroid. — The tropical 

 year will not continue indefinitely to diminish in length. When it has 

 once attained a certain minimum value, it will then increase, until finally 

 having attained an extreme value in the opposite direction, it will again 

 begin to diminish, and thus it will perpetually oscillate between certain 

 fixed limits. Laplace found that the extent to which the tropical year 

 is liable to vary from this cause amounts to 38 seconds. If it were not 

 for the effect produced upon the inclination of the equator to the eclip- 

 tic by the mean disturbing action of the sun and moon upon the ter- 

 restrial si)heroid, the extent of variation would amount to 1C2 seconds. 



Motion of the perihelion of the terrestrial orbit. — The major axis of 



