T', the time of the middle of the central eclipse for the earth in general; T―t"+t, will be the time of beginning; and T—t" + t', will be the time of end. 2. To the value of H at the time T, found by the last problem, add, (T' — T) .15°, and the sum will be the value of H' at the time T'. From this value of H', and to it, subtract and add, § (t—t). 15°, and the remainder and sum will be the values of H' at the beginning and end of the central eclipse. 3. To sin (N+F+ 180°), add cos D', and the sum will be the sine of the geocentric latitude ', not exceeding 90°, and north or south, acccording as the sine is affirmative or negative. To p', add the reduction of latitude, taken from tab. XVI, with ' as the argument, and the sum will be p, the latitude of the place at which the eclipse begins to be central. To cot. (N + F), add Ar. Co. sin. D' and the sum will be the tangent of an hour angle H, less than 180°, and affirmative when (N +F) is in the first or fourth quadrant, but negative when it is in the second or third. When H is to be negative, it must be taken greater than 90°, if its tangent is affirmative, but less than 90° if the tangent is negative. Subtract H from the value of H' at the beginning of the central eclipse, and the result will be λ, the longitude of the place at which the eclipse begins to be central. It will be west if affirmative, but east, if negative. Using (NF) instead of (N + F), and taking the value of H' at the end of the central eclipse, we find in like manner, the latitude and longitude of the place at which the eclipse ceases to be central. 4. To sin D', add tang N, and the sum will be the tangent of an arc M' not exceeding 180°, and with the same sign as D'. When M' is to be negative, it must be taken greater than 90°, if its tangent is affirmative, but less than 90°, if the tangent is negative. To sin D', add cot N, and the sum will be the tangent of an arc N', less than 90°, and with the same sign as its tangent. Then take S = M'N'. Find log. B' = tang D'Ar. Co. sin M' log. C'cos F + Ar. Co. cos D' + Ar. Co. sin N log. E' cos D' + sin N + cos N + log. of 15 + Ar. Co. log. of p'. log. G' sin N+ sin N + Ar. Co. cos N' + log. of 15+ Ar. Co. log. of p'. 5. Add M' to the value of H' at the middle of the central eclipse, and call the sum H". Add M' to the value of H at the beginning of the central eclipse and call the sum, L'; also add M' to the value of H at the end of the central eclipse and call the sum, L". The quantities found in the last article and this, may all be regarded as constant. 6. Let L be the value of (H+ M'), at any time during the central eclipse. Then assuming for L any value at pleasure within its limits L and L" to sin L, add log. B', and the sum will be the tangent of an arc B, not exceeding 90°, and with the same sign as its tangent. To cos B, add log. C', and the sum will be the sine of an arc C less than 90° and with the same sign as its sine. Then will B + C be the value of ' the geocentric latitude of the place at which the eclipse is central when L has its assumed value. From ' we find as in Art. 3. Add together sin ' and log. E' and the sum will be the logarithm of an arc E, in degrees and decimals of a degree. Add together cos p', sin (LS), and log. G', and the sum will be the logarithm of another arc G. Then will H" +E+ GL, be λ, the longitude of the place. We may thus by assuming a series of values for L, within its limits L' and L", find a series of places at which the eclipse will be central. Note 1. When two assumed values of L, differ only in the sign, the corresponding values of B, will differ only in the sign, and the values of C will be precisely the same. When two assumed values of L, are supplements of each other, the latitudes and values of E, will be the same for each. By attention to these circumstances, the computation of a series of places, may be considerably shortened. 2. If we subtract M' from any assumed value of L, the remainder will be the hour angle at the place at which the eclipse is then central. EXAMPLE. Required the times and places at which the eclipse of May 1836, began and ceased to be central; and also a series of other places at which it was central. d. h. m. From the calculation of the last problem, we have T = 15 2 0, D' 18° 57' 7, and H' = +30° 59′; also dividing the values of p g at the time T, and the values of p' and q', by 10000, we have, p = and By assuming for L various other values between its limits L' and L", the latitudes and longitudes of a series of places at which the eclipse will be central, as given in the following table, may easily be found. The computation of a part of these may serve as an exercise for the student. The value of a quantity at three consecutive whole hours, T — 1, T, and T+1, being given, to find its value at an intermediate time T, and its hourly variation at that time. Attending to the signs, subtract the value of the quantity at the time T-1, from its value at the time T; and its value at the time T, from its |