60 



KNOWLEDGE. 



[March, 1903. 



the determination of the Moon's place, and consequently 

 did not detect the Variation, since the latter vanishes (as 

 regards her direction) at both New and Full Moons. 



'I'he undisturbed path of the Moon round the Earth is 

 not, however, a circle, but an ellipse with ecceutricity y'g. 



This produces the elliptic inequalities in her distance 

 and direction. At perigee and apogee her direc^tion is the 

 same as that of the Mean Moon, but her distance is 

 ] 3,200 miles less or gi-eater than the mean distance, 

 238,818 miles. One quarter of a i evolution after perigee 

 she is at her mean distance, but (!■ lOj' in advance of the 

 Mean Moon, while one quarter of a revolution before 

 perigee she is an equal amount behind it. 



The first effect of the Sun's action on this elliptic motion 

 is to malie the perigee move round the Earth in a period 

 of 8 85 years, so that an anomalistic month (from perigee 

 to perigee) is slightly longer than a sidereal one. But in 

 addition to this continuous effect there is a periodic effect, 

 known as the Evection, which pi-oduees a large oscillation, 

 both in the eccentricity of her orbit and in the direction 

 of her perigee. When perigee occurs at New or Full 

 Moon the eccentricity rises from UOo-S to 00G6, while 

 when ])erigee occurs at First or Last Quarter it falls to 

 0-044. 



The perigee point has a mean annual motion of 40§", 

 but the Evection makes its motion oscillatory. At all 

 the four quarters the directions of true and mean perigees 

 coincide. When perigee occurs at the 1st or 5th octant 

 (vide the " variation " diagram) the perigee point lags 

 some 25*^ behind its mean position, while at the 3rd or 

 7th octant it is an equal amount ahead of it. At these 

 times the eccentricity of the orbit has its mean value 0-055. 



Owing to the Earth's orbit round the Sun not being 

 circular, the disturbing action of the Sun on the Moon 

 undergoes an annual variation. This produces the " annual 

 equation " of the Moon's longitude ; she is at her mean 

 place about January 1st and July 1st. some 11' behind it 

 about April 1st, and an equal amount ahead of it about 

 October 1st. 



Some readers may be glad to have the means of com- 

 puting with tolerable accuracy the Moon's longitude, 

 latitude, and parallax for any epoch. Thus questions 

 sometimes arise as to the date of New Moon in some 

 distant year, or the position of the Moon after some battle 

 or other historical event, and one has not always books of 

 reference at hand to give the information. The notation 

 in the following formuhe is that of Delaunay, but the 

 mmierical values of I, V, F, D, are due to Newcomb. 



We first find the interval in days between the ejioch 

 required and 1900, January 1, Greenwich noon. This 

 interval is called n, and is positive when the epoch is after 

 1900. AVe ne.\t calculate the values of /, /', F, D, from 

 the following formuke : — 



/ = Moon's Mean Anomaly = 309°174 + 13"-0G4993 n. 



V = Sun's Mean Anomaly = 359°-4t)5 + 0°-9S5600 ». 



r= Moon's Mean Longitude measured from Asc. 

 Node = 24=-482 + 13°-22935l «. 



D = Difference of Mean Longitude of Moon and Sun = 

 2°-930 + 12°-190750 ». 



Theu the following formula gives us the Moon's 

 Longitude : — 



Longitude = 283°-614 + 1.3°'l 76395 >i - 0°-186 sin. /' 

 + 6°-289 sin. I 

 + 0°-04l sin. {I - V) - 0-030 sin. (l + l') + 



0°-2l4 sin. 2 / 

 - 0''-114 sin. 2 F + 0°-65S sin. 2 D + 0=046 



sin. (2D -?') 

 + 0°-053 sin. (2 D + /) + l=-274 sin. (2 D - 



+ 0'-057 sin. (2 D - / - T) 

 + 0-059 sin. (2 D - 2 /) - 0°'035 sin. D. 



For the smaller tenns a very rough computation of the 

 angles will sufiice. The error of the deduced longitude 

 will seldom exceed 3'. For distant epochs we may apply 

 the following correction to the longitude for secular 

 acceleration (taken as 8" per century). 



In the above equation the term with argument ?' is the 

 " annual equation, ' that with / the " elliptic inequality" 

 with 2 F the " reduction," with 2 D the " variation," with 

 2 B-l the "Evection," with D the "Parallactic In- 

 equality." 



The "Moon's Latitude = 5°-I28 sin. F -|- 0=281 sin. 

 (F + I) - 0°-278 sin. (F - I) + 0°-033 sin. (2 D -f F) 

 -f- 0=055 sin. (2 D + F - ?) + 0°-173 sin. (2 D - F) 

 4- 0°-046 sin. (2 D - F - /). 



The formula for the Parallax is very simple ; the follow- 

 ing expression will give it to within a few seconds of arc : 



Parallax = 67'-04 + 3'-ll cos. I + 0'-l7 cos. 2 / -f 0'-47 

 cos. 2 D + 0'-57 cos. (2 D - ?)• 



The semidiameter may be found by taking .j^ of the 

 Parallax and then subtracting I", 



The term in the latitude with argument 2 D — F is 

 strikingly similar to the Evection. It has the effect of 

 making the inclination of her orbit to the ecliptic a 

 maximum (5" 18') when New Moon or Full Moon occur 

 at a node, while it is a minimum (5"^ 0') when First or 

 Last Quarter t)ccurs at a node. Thus the inclination is at 

 a maximum for all eclipses, and in consequence we get 

 rather fewer eclipses than we otherwise should. 



The motion of the node is also oscillatory as that 

 of the perigee is, but to a much smaller extent. Thus the 

 True Node coincides with the Mean Node when Nodal 

 passage occurs at any of the 4 quarters ; it is retrograding 

 at double the Mean rate when the node is passed at 

 quadrature, while it practically stands still when the node 

 is passed at syzygies. 



When the node is passed at the 1st or 5th octant the 

 longitude of the True Node is some 2' less than that of 

 the Mean Node, while at the 3rd or 7th octant it is an 

 equal amount greater than it. 



The method of expressing the perturbations as the sums 

 of series of sines and cosines of uniformly varying angles 

 is that which is emjiloyed in all the lunar and planetary 

 tables. When such tables have once been constructed, the 

 computation of the place of the Moon or planet at any 

 time is merely a matter of simple arithmetic. 



In the above description of the Moon's True path round 

 the Earth, no attempt has been made to explain thereascm 

 of the various inequalities ; it is not indeed easy to get far 

 in the theory of the subject without the use of analytical 

 methods. But in a difficult problem like the Lunar Theory 

 I believe the best method is to commence with an accurate 

 arithmetical conception of the nature of the Moon's 

 motion, after which theoretical researches can be pursued 

 with much ijreater interest and facility. 



