ON THE MAGNITUDE OF THE SOLAR SYSTEM. 101 



talk about the advantages to be obtained by observing a small, well- 

 defined crater instead of the moon's limb, astronomers have hitherto 

 found it impracticable to use anything but the limb, and the disadvan- 

 tage of doing so, as compared with observing a star, is still further 

 increased by the circumstance that in general only one limb can be seen 

 at a time, the other being shrouded in darkness. If both limbs could 

 always be observed we should then have a uniform system of data for 

 determining the ])lace of the center, but under existing circumstances 

 we are compelled to make our observations half upon <>ne limb and half 

 upon the other, and thus they involve all the systematic errors which 

 may arise from the conditions under which these limbs are observed, and 

 all the uncertainty which attaches to irradiation, personal eipiatiou, 

 and our defective knowledge of the moon's semidiameter. 



Our fifth gravitational relation is that which exists between the solar 

 parallax, the lunar parallax, the moon's mass, and the earth's lunar ine- 

 quality. Strictly speaking the moon does not revolve around the earth's 

 center, but both bodies revolve around the common center of gravity 

 of the two. In consequence of that an irregularity arises in the earth's 

 orbital velocity around the sun, the common center of gravity moving 

 in accordance with the laws of elliptic motion, while the earth, on 

 account of its revolution around that center, undergoes an alternate 

 acceleration and retardation which has for its period a lunar month, 

 and is called the lunar inequality of the earth's motion. We perceive 

 this inequality as an oscillation superposed on the elliptic motion of 

 the sun, and its semiamplitude is the measure of tlie angle subtended 

 at the sun by the interval between the center of the earth and the 

 common center of gravity of the earth and moon. Just as an astron- 

 omer on the moon might use the radius of her orbit around the earth 

 as a base for measuring her distance from the sun, so we may use this 

 interval for the same purjiose. We find its length in miles from the 

 equatorial semidiameter of the earth, the moon's parallax, and the 

 moon's mass, and thus we have all the data for determining the solar 

 parallax from the inequality in question. In view of the great diffi- 

 culty which has been experienced in measuring the solar parallax itself, 

 it may be asked, W^hy we should attempt to deal with the parallactic 

 inequality, which is about 26 per cent smaller "? The answer is, liecause 

 the latter is derived from differences of the sun's right ascension, 

 which are furnished by the principal observatories in vast numbers, 

 and should give very accurate results on account of their being made 

 by methods which insure freedom from constant errors. Fevertlieless, 

 the sun is not so well adapted for precise observations as the stars, 

 and Dr. Gill has recently found that heliometer measurements upon 

 asteroids which approach very near to the earth yield values of the 

 parallactic inequality superior to those obtained from right ascensions 

 of the sun. 



Our sixth gravitational relation is that which exists between the 

 moon's parallax and the constants of precession and nutation. Every 



