238 TT r . Harkness — Magnitude of the Solar System. 



Our third gravitational relation, to wit : that existing between 

 the solar parallax, the solar attractive force and the masses of 

 the earth and moon, is analogous to the relation existing be- 

 tween the moon's mass and parallax and the force of gravity 

 at the earth's surface, but it can not be applied in exactly the 

 same way on account of our inability to swing a pendulum 

 on the sun. We are therefore compelled to adopt some other 

 method of determining the sun's attractive force, and the most 

 available is that which consists in observing the perturbative 

 action of the earth and moon upon our nearest planetary neigh- 

 bors, Yenus and Mars. From this action the law of gravita- 

 tion enables us to determine the ratio of the sun's mass to the 

 combined masses of the earth and moon, and then the relation 

 in question furnishes a means of comparing the masses so found 

 with trigonometrical determinations of the solar parallax. 

 Thus it appears that notwithstanding necessary differences in 

 the methods of procedure, the analogy between the second and 

 third gravitational relations holds not only with respect to their 

 theoretical basis, but also in their practical application, the one 

 being used to determine the relation between the mass of the 

 moon and its distance from the earth, and the other to deter- 

 mine the relation between the combined masses of the earth 

 and moon and their distance from the sun. 



Our fourth gravitational relation deals with the connection 

 between the solar parallax, the lunar parallax, the moon's mass 

 and the moon's parallactic inequality. The important quanti- 

 ties are here the solar parallax and the moon's parallactic 

 inequality, and although the derivation of the complete expres- 

 sion for the connection between them is a little complicated, 

 there is no difficulty in getting a general notion of the forces 

 involved. As the moon moves around the earth she is alter- 

 nately without and within the earth's orbit. When she is 

 without, the sun's attraction on her acts with that of the earth ; 

 when she is within, the two attractions act in opposite direc- 

 tions. Thus in effect the centripetal force holding the moon 

 to the earth is alternately increased and diminished, with the 

 result of elongating the moon's orbit towards the sun and com- 

 pressing it on the opposite side. As the variation of the cen- 

 tripetal force is not great, the change of form of the orbit 

 is small, nevertheless the summation of the minute alterations 

 thereby produced in the moon's orbital velocity suffices to put 

 her sometimes ahead, and sometimes behind her mean place to 

 an extent which oscillates from a maximum to a minimum as 

 the earth passes from perihelion to aphelion, and averages 

 about 125 seconds of arc. This perturbation of the moon 

 is known as the parallactic inequality because it depends on the 

 earth's distance from the sun, and can therefore be expressed 



