LUNAR ECLIPSKS.] 



ASTRONOMY. 



975 



exactly the same situation. This exact return cannot 

 happen, but is sufficiently approximate to permit us to 

 foretel eclipses with a remarkable degree of accuracy ; and 

 Dr. Halley found that if the period of 18 years, 10 days, 

 7 hours, 43j minutes, were added to the middle of the 

 time of any eclipse, the return of the corresponding one 

 might be predicted within Ih. 30m. If only four leap- 

 years occur in the interval, the period would be enlarged 

 to 18y. lid. 7h. 43Jm. It may, however, happen that 

 when the eclipse is very slight, or an ajipulse occurs, that 

 at the succeeding period of 18y. lid. it will not be ob- 

 served. Since the position of the heavenly bodies at any 

 epoch, as well as the laws which govern their motions, 

 are at present equally well known, the eclipses of the sun 

 and moon are now rigorously calculated, although, to 

 save time and trouble, the above period of 38y. lid. (the 

 period of 621y. 3h. 3m. would be still more exact) is still 

 made use of. The Nautical Almanac contains the exact 

 positions of the sun and moon in relation to the earth at 

 any given time ; and it is from these data that we shall 

 determine the beginning and end of an eclipse at any 

 given time. 



In order to calculate an eclipse of the moon, the follow- 

 ing data are necessary, and the manner in which they 

 are usd will be seen by the diagram (Fig. 126) and ex- 

 Fig. 156. 



planation. Suppose the moon's place at opposition to be 

 at b, the centre of the earth's shadow being at a at the 

 game time, the latter passing through the space T a, the 

 path of the ecliptic, whilst the moon's centre describes 

 the line L b, the path of the moon's motion (which m.iy 

 be regarded as straight lines, as well as those of L c, a, b) : 

 Let m = moon's horary motion in longitude. 



n moon's horary motion in latitude. 



i = sun's horary motion in longitude. 



X =- moon's latitude when in opposition at 6. 



t time taken by moon to pass from L to 6. 



d = distance from T to L. 



The moon's motion in longitude will be o c in the in- 

 terval it takes to pass from L to 6 = m t. Its motion in 

 latitude, during the same time, or L e a b n t. The 

 sun's motion in longitude, or the distance T a = i 1 in the 

 Dame manner. Then 



Lc = o6-(-nt = X-)-n< and T e = ac Ta=mt 1 1 

 tl* = L T 1 = Lc* + Tc 1 = (X + TV t) s + (mt - tt)\ 

 Expanding this expression, we obtain a quadratic 

 equation of which t is the unknown quantity (the others 

 being derived from the above data), and will depend on 

 the value given to d. Such values may be given to (i as 

 correspond to the phases of the eclipse, and the interval 

 between the time of opposition, and the occurrence of 

 the phases will thus readily be obtained, the time of 

 opposition being known, from the position of the sun and 

 moon an given by the tables. Arranging them, we 

 obtain 



<P V - t 1 [(m-) J +- 2 ] + 2tXn. 



sin.*0 n 2 



Substituting tan. 1 <= , - : ^-f. i or -. - r,, we 

 1 sm. 2 (riv ) 2 ' 



obtain, instead of the above, 



n 2 P 4 2 X n sin. 1 t = (d? - X 1 ) sin. 1 ; or, completing 

 the quadratic, 



nH* + 2 Xn gin. t + X 1 sin. * = (<P X) sin. *0 + 

 X 2 nin. *0 



-sin. 2 (d~X*) (1 -sin.* ) 



.*. I - - ( X sin. 1 + sin. 6 ,/d? X 2 cos. 2 0. 

 n 



And from this expression the values of the times for any 

 values of d may be obtained. For the determination of 

 the time, *, at which the moon immerges into the earth's 

 penumbra, d = moon's horizontal parallax + sun's hori- 

 zontal parallax + the sun's semi-diameter + moon's 



semi-diameter = P + p + +> the first value of t 



giving the instant of immersion, the second that of 

 emersion. At the time of immersion in the umbra 



D 

 d = P -|- p + 2 ~n' At the period when the whole 



disc has entered the shadow d = sum of the parallaxes 



sum of the semi-diameters, or P + p s ^ The 



& 



time at which the middle of the eclipse occurs (or the 

 greatest phase) is when the two halves of t are equal, or 



when d* X 3 cos. s = 0, and t = - : In this case 



the distance d = X cos. 0. When the latter term is 

 known, the magnitude of the eclipse may be determined. 

 Some part must be eclipsed when X cos. in less than. 



the distance P + p -| --- ^ as the latter quantity 

 2 2 



shows when the moon's limb just touches the shadow, 

 and the portion of diameter eclipsed is ( P -)- p -f- o -5 ] 



X cos. 0. The portion of the diameter of the moon 

 which is not eclipsed will, therefore, be X cos. + 5- + 

 2 P p. The eclipse will be exactly total when this 



is nothing, and will be more than total when it is nega- 

 tive. The part eclipsed may be expressed either in digits 

 or twelfths of the lunar diameter (<S), or in decimal parts, 

 the moon being taken for unity. 



Example. Partial eclipse of the moon, April 20th, 

 1856. 



By the tables of the sun and moon, the time of oppo- 

 sition is at 9h. 13m. -0 morning, Greoenwich mean time. 



Moon's lat. at time of opposition X=33 18 



,, horary motion in lat. . n= 2 4i> lat. increasing. 



horary motion in long. . m=30 8 

 Sun's horary motion in long. . = 2 27 

 Moon's apparent diameter . . 5 = 29 44 



,, horizontal parallax . . P=54 33 

 Sun's apparent diameter . . D = 31 52 



,, horizontal parallax . . p= 9 







5' 43' 10-. 



The middle of the eclipse, or 

 Xsin. 2 Q = + 1998^ ^ , ^ 



Tlierefore the midddle of the eclipse will occur at 

 9h. 6m. '4 morning, reckoning from opposition. The 

 times at which it entered the shadow and emerged from 

 it are calculated by finding the value of d at those 

 times : 



d = jj g ; + p + P = 53' 38", and adding the one- 

 sixtieth part of f P + p 5- j for the effect of the earth's 



the expression 



atmosphere, d = 54' 17*. Introducing this value into 

 XBJn.'0 _^ sin. J (d 2 X 2 cos. 2 0) 



n 

 we obtain for the two values of t : 



End of eclipse = 716 + 1 32-68 = 1 39-84 

 Beginning = 716 1 32" 68 - 1 25-52 



