Aa" = sa' + ▲oa' = sa + 24oa + s3a Aa"" = ▲a" + ▲2a" = ▲a + 34oa + 3▲3a + s1a Δα"!!! = Aa"" + A3a"" = ▲a + 442a + 643a + 4▲1a + ▲3a. Hence, since a' = a + ▲a, a" = a' + sa', &c., we have, a' = a + sa a" = a + 2▲a + s'a a"" = a + 34a + 3A3a + ▲3a a""" = a + 4▲a + 64oa + 4▲3a + s1a a"""" = a + 5Aa + 10A a + 10A3a + 5s1a + s3a. The co-efficients of the terms in the values of a, a', a", &c., which are the values of x at the times T, T+ 1, T + 2, &c., are, therefore, the same as the co-efficients of the terms in the first, second, third, &c, powers of a binomial. Hence, if we take t for an interval of time, expressed in hours, or hours and parts of an hour, we shall have for the value of x at the time T + 1, 60. The process of interpolation can only be employed with advantage when the differences of some order, not very high, become equal, or very nearly equal. If we suppose the fourth differences to be equal, then the fifth and higher differences must 0. Hence, we would have in this case, for the value of x at the time T +t, the rigorous formula, t (t − 1) s'a + If the differences of the fifth and higher orders, though not absolutely = 0, are very small, the formula will still be very nearly accurate for values of t not exceeding, or not much exceeding 4hrs. For, in this case the co-efficients of the terms in which these higher differences enter, being each less than a unit, the terms omitted will be very small. 61. Another Formula. We may obtain a formula, that is frequently more convenient than those above, by taking the times and values of x, as in the following table. Put sa + sɑ, = 2b, s2α, = c, s3a, + s3α, Then, since half the difference of two quantities, added to half the sum, gives the greater quantity, we have, sab+c, and s3a, = d + e. Hence, assuming the fourth differences to be equal, s'a e, s'a = s3a, + e=d+e, and s'a s2a, + s3a, = c + d + { e. Substituting these values of sa, s3a, s3a, and Ata, in the formula (B), it becomes, = Performing the multiplications indicated, and reducing, we have for the value x, at the time T + t, The interval t, may be taken of any value between 2 and +2. - 62. The hourly variation of a quantity at any time, is the variation or change in its value which would take place in an hour, if the rate of change at that time were to be the same, throughout the hour. The average hourly variation of a quantity between two times T and T+t, is that hourly variation which, if continued during the interval t, would produce the change in the value of the quantity, that actually takes place during this interval. 63. Average hourly variation. It follows from the definition, that the difference between the values of a quantity at the times T and T + t, divided by the interval t, must give the average hourly variation during the interval. Hence, if we put x' =this average hourly variation, we have, from the formula (D), The average hourly variation or the values of x' for the whole hours, are, 64. Hourly variation. If in the formula (D) we put t + instead of 1, we shall have for the value of x at the time T+t+t', 12 14 x = a + 1 (b − jd) + —— (c — me) + 1d + £e + 1 (b — jd) t 2 4 t t + 6t t'2 + 4 tt's +1+ 24 t' 6 e. 6 24 From this value of x, subtracting its value at the time T + t, (D), and dividing the remainder by t, we find the average hourly variation between the times T+t, and T + t + t', to be, Now, it is evident that the smaller the interval t', is, the nearer will this average hourly variation approach to the hourly variation at the time T +t. Hence, if we now take x' to stand for the hourly variation at the time T + 1, we have by taking t = 0, in the above expression, The hourly variations or the values of x' at the whole hours, are, (G) (H) INVESTIGATION OF FORMULE FOR COMPUTING SOLAR ECLIPSES, OCCULTATIONS, AND TRANSITS. 66. Let O, Fig 64, be the centre of the earth, A a place on its surface, S the centre of the sun, M that of the moon, and S', M', A', s, and m, the points in which OS, OM, OA, AS, and AM, produced, meet the celestial sphere. Then will S' and M' be the true places of the sun and moon, s and m; their apparent places, and A', the geocentric zenith of the place A. Let a be the zenith of the place A, OZ a straight line parallel to MS, meeting the celestial sphere in Z, EQ the equator, E the vernal equinox, P the north pole of the equator, and PB,PC, PF, and PK, declination circles through Z, S', M', and a and A. Also, let BX and ZY be each a quadrant. Then, since BX is a quadrant, X is the pole of the declination circle YB; and, consequently OX is perpendicular to OY and OZ. Also, since ZY is a quadrant, OY is perpendicular to OZ. Hence OX, OY, and OZ form a system of rectangular axes, having their origin at O, the centre of the earth, and having the axis OZ parallel to MS, the line joining the centres of the moon and sun. 67. Taking the equatorial radius of the earth = 1, let, x, y, z, be the co-ordinates of M, Also let, ę = OA = distance of place A from earth's centre, G sun's mean distance, = MS distance between the centres of the moon and sun, A = EF = right ascension of the moon, = A' EC = α= EB = μ = EK = FM' = declination of the moon, D = D' d = BZ ? = Ka point a, or geogr. Lat. of A, = moon's equatorial horizontal parallax, 68. To find the values of a, d, and g. Let EX' be a quadrant. Then OX', OP, and OE will evidently be another system of rectangular axes, having the same origin as the former system. On OZ, take OL = MS, and let MG, SH, and LI be perpendicular to EOQ, the plane of the equator, meeting it in G, H, and I. Also let GU, HV, and IW be perpendicular to the axis OE, and let GN be parallel to it. As MS and OL are parallel and equal, their projections GH and OI are parallel and equal. Hence, as GN is parallel to OW, the right angle triangles GNH and OWI, are equal, and we have OW-GN = UV = OV-OU. Consequently that ordinate of the point L, which is parallel to the axis OE, is equal to the difference between the ordinates of S and M, parallel to the same axis. The same relation must evidently have place for the ordinates parallel to the axes OX' and OP. Hence, if we put a, a', and a" for the ordinates of M, S, and L, parallel to OX'; B, B', and B", for those parallel to OP; aud y, y', and y", for those parallel to OE, we shall have, a" a' = B'- B; and y" — y. B" = = OL cos BOZ = = OL sin BOZ G sin d; = The co-ordinates of S = y= G cos d cos a R' cos D' cos A' R cos D cos A and a" = -a; G cos d; Also B"IW OI sin EOB G cos d = = G cos d cos a, = and M will evidently have similar expres B" G sin d a" G cos d sin a =R' cos D' sin A' a = R cos D sin A cos A' R cos D cos A G cos d sin a = R' cos D' sin A' R cos D sin A G sin d -R' sin D'-R sin D. Multiplying the first of these last three equations by cos A', and the second by sin A', and adding the products; and multiplying the first, by sin A', and the second by cos A', and subtracting the first product from the second, we obtain, |