The equation of time arises from two causes, first, the variable motion of the sun in the orbit, and secondly, from the obliquity of the ecliptic to the equator. ally before the real sun, and the former will be later in coming to the meridian; i. e. calling t' that part of the equation due to the variable motion, t' will be subtractive from apogee to perigee. Again from perigee to apogee, the mean sun is behind the real sun, or comes on the meridian first, hence t' from perigee to apogee is additive. Call t that part of the equation of time due to the obliquity. Letry and r z represent halves of the equator and ecliptic respectively. P is the pole of the equator. Let us now suppose another fictitious body S2 to move uniformly in the equator, completing its revolution in the same time with that of the revolution of the real sun. = Let S1 be the position of S1 in the ecliptic at any time. Take ry 90°; draw (P r,) P 81 s a secondary to the equator through $1, PZY another secondary cutting the ecliptic in Z. Now by Napier's Rules Cos S1 s = Coss Sin S1 r s = Sin Cos r s' if w = obliquity S1 rs. ifrs Cos = 90°, Cos r s is positive, and Sin w is also positive, .. S1 s is positive .. Sls 90°, .. r S s ▲ r s $1 or $1 > r s or S1 is behind S2 (the equator mean sun) or S1 comes on the meridian first, .. t is subtractive from an equinox to solstice, when Ts= =ry, Cos r so,.. Cos r Sloor ↑ S1 s = 90°, i. e. r zry, or both S1 and S2 come on the meridian at once. if r s > 90°, Coss is negative. .. Cos or S1 S1 s is negative, or S1 s > 90° s, or S2 comes on the meridian first, .. from solThese being premised we proceed to (e) is 0 four times in the year. stice to equinox t is additive. shew that the equation of time In this figure, p and a are the perigee and apogee; E the earth; V, A the vernal and autumnal equinoxes; S and W the summer and winter solstices. M is the place where t attains its maximum value, which is greater than even the maximum value of t' in consequence of the change of sign of e between (1) .... W and p, e must have been nothing in the interval. From P to V, et at V, e t' From V to S, e = ť at M, t > t', (2) .. Hence e .... .. at M, e(tt) is negative o, between V and M. at S, e t' (3).... Hence, eo between M and S (4) .... Hence e= t', at W, e = o, between a and A. From A to W E -t', .. e is not o between A and W. JUDONAUTH DOSS, Calculation of an Eclipse of the Sun, October 9th, 1847. To find the time of true Ecliptic Conjunction. To do this we must see when the true longitudes of the Sun and Moon are perfectly equal; and as the apparent longitude of the Sun is given in the Ephemeris we must first correct it for aberration. Sun's longitude. Having first found by common proportion that the ecliptic conjunction happens sometime between October 8, 21 hours, and October 8 21h 10m we proceed to find it more accurately by applying the second differences the arguments of which we have determined above. 12h 9h: 5° 53′ 43′′.1: 4° 25′ 17′′.32 12h 9h 10m :: 5° 53′ 43′′.1: 4° 30′ 12′′.09 O's motion in long. in 10m = 24.729 ... 4' 54".87 24".729 = 4' 30".141 is the relative motion of the Moon in long. in 10 minutes. Now 195° 28′ 53′′.15 — 195° 27′ 41′′.211 = 1' 11".939 is the difference of the longitudes of the and D at 21h 10m; therefore to determine what time elapsed after the true ecliptic conjunction during which by the motion of both or the relative motion of the Moon the difference of longitude becomes 1' 11".939, we have immediately the proportion 4′ 30′′.141 : 10m :: 1' 11".939: 2m 39s.78 before To verify this and at the same time to find the long. of the Sun and Moon at the time of 6 12h 9h 7m 20s.22: 4° 28′ 53′′.594 2.075 4 28 51.519 190 58 43.1 and for the Sun 24h: 21h 7m 20$.22 :: 59′ 20′′.906: 52′ 13′′.936 .1 52 13.836 194 35 20.790 195° 27′ 34′′.619 195° 27′ 34′′.626 7 This differing only by .007 or 1000 part of a second. 12h 9h 7m 20s.22 :: 32′ 41′′.3 : 24' 50'.96 To find the relative motion in orbit and its inclination to the ecliptic. Take AB =4' 30".141 = 270".141 motion in Longitude |