534 



NA TURE 



[September 27, 1S94 



a iraverse survey along the old Post Road from New Vork to 

 Albany, and finally, lo determine the latitude of some point in 

 Albany. The traverse survey should surely be correct to one 

 part in three hundred, and .as the distance between the two 

 cities is about two degrees, the difl'erence of latitude mii;ht be 

 determined to about the same percent.ige of accuracy. In that 

 way we would tind the length cf two degrees of latitude to be 

 about 13S miles, whence the earth's radius would be 3953 

 miles. It would then only remain to observe the time occupied 

 by the moon in making a sidereal revolution around the earth, 

 or, in other words, the time which she occupies in moving from 

 any given star back 10 the same star again. By noting that to 

 within one-quarter of her own diameter we should soon tind 

 tlut the time of a revolution is about 27'32 days, and multiply- 

 ing that by the number of seconds in a day, viz. 86,400, we 

 would have for the lengih of the sidereal month 2,360,000 

 seconds. With these data the compulation would stand as 

 follows: — The radius of the earth, 3953 miles, multiplied by 

 the length of a sidereal month, 2,360,000 seconds, and the 

 pioduct squared, gives 87,060.000,000,000,000,000. Multiply- 

 ing that by one-fourth ol the lengih of the seconds pendulum, 

 vii. I/64S0 of a mile, and extracting the cube root of the pro- 

 duct, »e would get 237,700 miles for the distance from the 

 earth to the moon, which is only about 850 miles less than the 

 truth, and certainly a remarkable result considering the crude- 

 ness of the instruments by which it might be obtained. 

 Nevertheless, when all the conditions are rigorously taken into 

 account, these data are 10 be regarded as determining the rela- 

 tion between the moon's mass and parallax rather than the 

 parallax itself. 



Our third gravitational relation, lo wit, that existing between 

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

 the earth and moon, is analogous lo the relation existing between 

 the moon's mass and )>arallax and the force of gravity at ihc 

 earth's surface, but it cannot be applied in exacily the same 

 way, on account of our inability lo 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 av.iilable 

 is that which consists in observing the perlurbative action ^of 

 the earth and moon upon our neaicsl planetary neighbours, 

 Venus and .Mars. Eiom this action the law of gravitation 

 enables us to determine the ratio of the sun's mass to the com- 

 bined masses of the eaith 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 relaiions holds nol 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 masse-; of the 

 earth and noon and their distance from the sun. 



Our fourth gravitational relation deals with the connection 

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

 and the moon's parallactic inequality. The important ([uantities 

 are here the solar parallax and the moon's parallactic inequality, 

 and alihough ihe derivation of ihe complete expression for the 

 connection between ihcm is a lillle complicated, there is no 

 diRicnlly in gelling a general notion of the forcej involved. .-Xs ' 

 the moon moves around Ihe earth she is niternateiy without and 

 wiihin Ihe earth's orbit. When she is without, the sun's .ittrac- 

 tioD on her acts with that of the earth ; when she is within, the 

 two attractions act in opposite directions. Thus in elTcct the cen- 

 trifugal force holding the moon tolhc earth is alternately increased 

 and diminished, wiih the result of elongating the moon's orbit 

 towards Ihc sun and compressing it on the opposite side. As the 

 variation of Ihe centrifugal force is not great, the change of 

 the form of the orbit is small, nevertheless the summation of 

 the minute alterations thereby produced in Ihe moon's orbilal 

 velocity .sutfices lo put her sometimes ahead, and sometimes 

 behind her mean place lo an extent which oscillates (rom a 

 maximum lo a minimum as Ihe earth passes from perihelion lo 

 aphelion, and averages about 125 seconds of arc. This per- 

 turbation of Ihc moon'n node i> known as Ihe parall.iclic 

 incijuality because It depends on Ihc earth's distance from the 

 •un, and can therefore be cxpresned in terms of the solar 

 parallax. Conversely, the solar parallax can be deduced from 

 the obscrvc'l value of the parallactic inequality, but unfortun- 

 ately there aie great practical difficulties in making the requisite 



NO. I3CO, VOL. 50] 



observations with a sufficient degree of accuracy. Xotwith- 

 standing the ever-recurring talk about the advantages to be 

 obtained by observing a small well-defined crater instead ol 

 the moon's limb, astronomers have hitherto found it impractic- 

 able to use anything but the limb, and the disadvantage of 

 doing so as compared with observing a star is still (uriher 

 increased by the circumstances that in general only one limb 

 can be seen at a lime, 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 place of the centre, 

 but under existing circumstances we are compelled to make our 

 observations half upon one 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, an<J 

 all the unceitainty which attaches to irradiation, personal 

 equation, and our defective knowledge of Ihe moon's semi- 

 diameter. 



Our fifth gravitational relation is that which exists between the 

 solar parallax, the lunar parallax, the moon's mass, and the earth's 

 lunar inequality. Strictly speaking, the moon does not revohe- 

 around the earth's centre, but botli bodies revolve around the 

 common centre of gravity of the two. In consequence of that 

 an irregularity arises in the earth's orbilal velocity around ihe 

 sun, the common centre of gravity moving in accordance with 

 the laws of elliptic motion, while the earth, on account of its 

 revolution around that centre, undergoes an alternate accelera- 

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

 is called the lunar inequality of the earth's motion. We per- 

 ceive this inequality as an oscillation superposed on the elliptic 

 motion of the sun, and its semi-amplitude is a measure of the 

 angle subtended at the sun by the interval between the centre 

 of the earth and the common centre of gravity of the carlh and 

 moon. lust as an astronomer 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 

 purpose. We find its lengih in miles from the equatorial semi- 

 diameter of the earth, tiie moon's p.irallax ami the moon's 

 mass, and thus we have all the data lor determining the solar 

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

 dilticuliy which has been expeiienced in measuring the solar 

 parallax itself, it may be asked why we should attempt to deal 

 with the parallactic inequality which is about Isventysix per 

 cent, smaller? The answer is, because the latter is derived 

 from dilfcrences of ihe sun's right ascension which are furnished 

 by the principal observ.itories in vast numbers, and should give 

 very accurate results on account of their being m.adc by methods 

 which insure freedom from constant errors. Nevertheless, the 

 sun is no! so well adapted for precise obsetvalions 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 ol ihc sun. 



Our sixth gravitational relation is that which exists between 

 the moon's parallax and the constants of precession and nuta- 

 tion. Every particle of the earth is attracted both by the sun 

 and by the moon, but in consequence of the polar llattening Ihe 

 resultant of these attractions passes a little to one side of the 

 earth's centre of gravity. Thus a couple is set up, which, by 

 ils action upon the rotating earth, causes the axis ihercol to 

 describe a surface which may be called a fluted cone, with in 

 apex at the earth's cenlre. A lop spinning with ils axis in- 

 clined describes a similar cone, except that the flutings are 

 absent, and the apex is at Ihe point upon which the spinning 

 occurs. For convenience of computation we resolve this action 

 into two components, and we name thai which i>ro(luces thtf 

 cone the luni-solar precession, and that which pioduces the 

 lluiings the nutation. In this phenomenon the part jilayed by 

 Itie sun is comparatively small, and by eliminating it we obtain 

 a relation between Ihe luni-solar precession, the nutation and 

 the moon's parallax, which can be used to verify and correa 

 the observed values of these quantities. 



In the preceding paragraph we have seen that the relation 

 between the quantities there considered depends largely upon 

 the flattening of ihe eaith, and thus we are led lo impure how 

 and with whal degree of .accuracy that is determined. There 

 are five methods, viz. one geodetic, one gravitational, and three 

 astronomical. The geodetic method depends upon measure- 

 ments of the length 01 a degree on various parts of the earth s 

 surface, and with Ihe data hitherto accumulated it has provesl 

 quite unsatisfactory. The gravitational method consists in de- 



