100 ON THE 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 between 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 toado])t 

 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 neighbors — 

 Venus and Mars. From this action the law of gravitation 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 

 ditferences in the methods of i^rocedure, the analogy between the sec- 

 ond 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 determine 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 i^arallax, the lunar parallax, tlie moon's mass, and the moon's 

 parallactic inequality. The important quantities are here the solar 

 parallax and the moon's parallactic inequality, and although the deriva- 

 tion of the conqdete expression for the connection between them is a 

 little complicated, there is no dititiculty 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 directions. Thus in effect the cen- 

 tripetal force holding the moon to the earth is alternately increased and 

 diminished, with the result of elongating the moon's orbit toward the 

 sun and conq)ressing it on the opposite side. As the variation of the 

 centripetal 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 i)lace to an extent which oscillates from a 

 maximum to a minimum, as the earth passes from perihelion to aphelion, 

 and averages about 125" of arc. This perturbation of the moon is 

 known as the parallactic inequality, because it dejiends on the earth's 

 distance from the sun, and can therefore be expressed in terms of the 

 solar parallax. Conversely, the solar parallax can be deduced from the 

 observed value of the jjarallactic inequality, but unfortunately there 

 are great practical difficulties in making the requisite observations with 

 a suliicieut degree of accuracy. Notwithstanding the ever-recurring 



