more convenient to regard the centre of the shadow as fixed at C, and to use the moon's relative motion in reference to this centre. Draw, therefore, M'm parallel and equal to C'C. Then, will mC be parallel and equal to M'C'. Hence, m is the moon's relative place at the time T + t, in reference to C, the fixed position of the centre.

= =

As CC' is the motion of the centre of the shadow during the interval t, it must, evidently, be equal to the apparent motion of the sun during the same time. Let M'D and md be drawn parallel to MC, and ME parallel to AB. Then Ee = M'm CC'. Consequently, Ee is equal to the sun's motion during the interval t; also ME is the moon's motion in longitude, and em = EM', is her motion in latitude during the same time. Hence, Me, the moon's relative motion in longitude during the interval t, is equal to the difference between her motion in longitude and that of the sun, and em, her relative motion in latitude, is equal to her real motion in latitude.

Let t' be a different interval of time, and let m' be the moon's relative place at the end of this interval. Then, since the motions are regarded as uniform (244), Me': Me: t': t, and e'm': em :: t't, or, Me': Me: : e'm' : em. Hence, m' must be in the straight line PQ, drawn through M and m. As, therefore, the moon's relative place moves along the line PQ, this line is called the moon's relative orbit.

247. Inclination of moon's relative orbit. Put I= the angle eMm the inclination of the moon's relative orbit. Then, expressing the interval t in the last article in hours and decimal parts, we have,

248. Moon's hourly motion in her relative orbit. Let n moon's hourly motion in her relative orbit. Then, we have,

But (247), Metp'. Hence, tn cos I tp', or,

249. Time of the middle of the eclipse. Let AB, Fig. 42, be the ecliptic, PQ the moon's relative orbit, and C and M the places of the centres of the earth's shadow and moon at the time of opposition or full moon. Also, let the circle KeLa, described about the centre C with a radius equal to S (226 Schol.). represent the section of the earth's shadow, at the moon. With the same centre and a radius equal to (S + 8) or h, let arcs be described, cutting the relative orbit in D and E; and let CN be perpendicular to the orbit PQ, cutting it in H. Then, will D and E be the moon's places at the beginning and end of the eclipse. Hence, as CH evidently bisects DE, and as the moon's motion is regarded as uniform during the eclipse, the point H must be the moon's place at the middle of the eclipse.

Let T' the time of the middle of the eclipse, and t the interval between T' and T, the time of full moon.

Then, MH tn. But, since CM is perpendicular to AB, and CH is perpendicular to PQ, the angle MCH is equal to the inclination of the relative orbit. Hence, from the right angled triangle CHM, we have MH = CM sin MCH = q sin I. Consequently,

The upper sign must be used when the latitude is increasing, and the lower when it is decreasing.

250. Beginning and end of the eclipse. From the triangle CHM, we have, CH CM cos MCH q cos I. Put,

B the time of the beginning of the eclipse,

E the time of the end,

t

the interval between the middle and beginning or end. Then we have, HD=tn; and from the right angled triangle CHD,

Therefore, BT' - t, and E T'+t, become known.

251. Times at which the eclipse begins and ceases to be total. With the centre C and a radius equal to S-8, or h', let arcs be described cutting the relative orbit in F and G. Then, will F and G be the moon's places at the beginning and end of the total eclipse. Put,

B'

the time the total eclipse begins,

E' the time it ends,

t = the interval between either of these and the time of the

When h' is less than CH or q cos I, the eclipse cannot be total.

252. Quantity of the eclipse. The quantity of an eclipse, either ́of the sun or moon, is usually expressed in twelfths of the diameter, which are called Digits. In a total eclipse of the moon, the quantity of the eclipse is denoted by the number of digits contained in the distance between the inner edge of the moon and the nearest opposite edge of the shadow. Thus, in the eclipse represented in the figure, the number of digits contained in LN expresses the quantity of the eclipse. eclipse. Then,

Let Q

the quantity of the

COMPUTATION OF SOLAR ECLIPSES.

253. Time of New Moon. The approximate time of new moon. being found, and also the sun's and moon's longitudes, &c., at that time (245), the difference between the longitudes will be the moon's distance from conjunction. Whence, by means of the hourly motions in longitude, the true time of new moon may be obtained.

254. General Eclipse. Taking h = x − x' + &' + ♪, (238), the times of the middle, beginning, and end of an eclipse of the sun, for the earth in general, may be found in the same manner as those of a lunar eclipse.

255. Eclipse for a given place. Although the calculation of an eclipse of the sun for the earth in general, is equally simple with that of a lunar eclipse, it is quite different when the computation is to be made for a given place. This is much more difficult and tedious. For, the circumstances of the eclipse at a given place depend on the apparent relative positions of the sun and moon, that is, on their relative positions as seen at the given place. It therefore becomes necessary to take notice of the effect of parallax in changing the apparent relative positions of the bodies. Referring to the appendix for a more full and minute investigation of the subject, we shall here only give a general view of a method by which the computation for a given place may be made.

!

As the sun's parallax is very small, and it is only the apparent relative places of the sun and moon that are required, we may, withcut material error, refer the whole effect of parallax to the moon; that is, we may regard the sun's true place as being his apparent. place, and then, in computing the moon's apparent place, use -x!, the difference of the moon's and sun's horizontal parallaxes, instead of, the moon's parallax.

Let T be time of the whole hour next preceding the approximate time of new moon, and for the time T, let the quantities mentioned in Article 245, be computed; and by means of the hourly motions, let the sun's longitude and the moon's longitude and latitude be found for the time T + 1hr. Then, using я instead of, let the moon's parallaxes in longitude and latitude

be computed for the times, T and T+ 1hr, and, thence, let her apparent longitudes and latitudes for these times be found. The difference between the moon's apparent longitude, at the times, T and T+1hr., will be her apparent hourly motion in longitude, and the difference between this and the sun's hourly motion will be the moon's apparent hourly motion relative to the sun, which may be called p'. The difference between the moon's apparent latitude, at the times, T and T + 1hr., will be q', the moon's apparent hourly motion in latitude. The difference between the sun's longitude at the time T, and the moon's apparent longitude at the same time, will be the moon's distance from apparent conjunction, at that time. Whence, from the value of p', the time of apparent conjunction may be found. The time, thus obtained, will however only be an approximate time, for the moon's apparent motions are not uniform. Let, now, the moon's apparent longitude and latitude be computed for the approximate time of the apparent conjunction, and the moon's distance from apparent conjunction at this time be found. This distance will, now, be very small. Hence, we may, by means of p' and q', find the true time of apparent conjunction, and the moon's apparent latitude q, at that time, very nearly. Also, with p' and q', we may, by Art. 247, find I, the inclination of the moon's apparent relative orbit.

Let AB, Fig. 43, be the ecliptic, and PQ the moon's apparent relative orbit. Then, SM being drawn perpendicular to AB and equal to q, the moon's apparent latitude at the time of apparent conjunction, S and M will be the place of the sun's and moon's centres, at that time. With the centre S and a radius equal to d', the sun's semi-diameter, let the circle ab be described to represent the sun's disc. Let SD and SE be each equal to '+8, the sum of the semi-diameters, of the sun and moon. Then will D and E be the moon's place, at the beginning and end of the eclipse. Hence, taking h='+d, the times of middle, beginning, and end, and the quantity of the eclipse, may be found, in exactly the same manner, as for an eclipse of the moon, except that, in finding the quantity, s' must be used in the denominator of the fraction instead of 8.

But, as the moon's apparent motions in longitude and latitude are not uniform, the times of beginning and end, thus found, are