414 LECTURE XLIII. 



revolving radius, until, at a certain point, which is called the lower apsis, 

 it becomes perpendicular to it : but if the central force increase in a greater 

 proportion than is necessary for the description of the ellipsis, the point 

 where the velocity prevails over it will be more remote than in the ellipsis ; 

 and this is expressed by saying that the apsis moves forwards. When, on v 

 the contrary, the force varies more slowly, the apsis has a retrograded 

 motion. Since, therefore, the force attracting the moon towards the earth, , 

 increases, on the whole, a little more rapidly than the square of the distance' 

 decreases, the apsides must have, on the whole, a direct motion. An4 a 

 similar theory is applicable to the mutual perturbations of the prinfary 

 planets. (Plate XXXIV. Fig. 488.) 



The secular acceleration of the moon's mean motion, which had long 

 presented a difficulty amounting almost to an exception, against the suffi- 

 ciency of the theory of gravitation, has at last been satisfactorily deduced 

 by Mr. Laplace from the effect of the gradual change of the eccentricity of 

 the earth's orbit, which is subject to a very slow periodical variation, and 

 which causes a difference in the magnitude of the sun's action on the lunar 

 revolution. 



The perfect coincidence of the period of the moon's rotation, with that 

 of a mean revolution, has been supposed to be in some degree an effect of 

 the attraction exerted by the earth on a prominent part, of her surface ; 

 there are, however, many reasons to doubt of the sufficiency of the expla- 

 nation. If the periods had originally been very nearly equal, we might 

 imagine that the motion of the earth would have produced a libration or 

 oscillation in the position of the moon, retaining it always within certain 

 limits with respect to the earth ; no libration is, however, observed, that 

 can be derived from any inequality in the moon's rotation ; and it has 

 very properly been suggested that the same attraction towards the earth 

 ought to have made the moon's axis precisely perpendicular to the plane of 

 her orbit, instead of being a little inclined to it. At the same time the 

 appearance of a similar coincidence, in the periods of the rotation and re- 

 volution of many other satellites, makes it probable that some general 

 cause must have existed, which has produced the same effect in so many 

 different cases. 



The orbits of the comets afford no very remarkable singularity in the 

 application of the laws of gravity, excepting the modifications which depend 

 on their near approach to the parabolic form, and the great disturbance 

 which their motions occasionally suffer, when they happen to pass through 

 the neighbourhood of any of the larger planets. The velocity of a comet 

 in its perihelion is such, that its square is twice as great as the square of 

 the velocity of a body revolving in a circle at the same distance. It was 

 determined by Halley and Clairaut, that the attractions of Jupiter and 

 Saturn would delay the return of the comet of 1759 about 618 days ; and 

 the prediction was accomplished within the probable limits that they had 

 assigned for the error of the calculation. The labours of Clairaut have 

 indeed in many respects improved the science of mathematical astronomy ;> 

 he was the first that obtained a complete determination of the effects of the 

 mutual actions of three gravitating bodies, disturbing each other's motions ; 



