2. Required to calculate, for the meridian of Philadelphia, the eclipse of the moon, on the 9th of April, 1838. PROBLEM XV. Given the Latitude of a place, to find the logarithms of p cos' and sin q', in which q' is the Geocentric Latitude and p the Radius of the Earth, for the given place. P h. m. 7 31/0 8 58 10 241 To the log. cosine and log. sine of the given latitude, add respectively, log. x and log. y, taken from table XVII., with the latitude as the argument, and the sums will be log. p cos q' and log. p sin p'. Note. When the logarithms are required to more than five decimal figures, they may be found by the formulæ (App. 51, A and B). EXAM. Required the logarithms of p cos p' and p sin p', for Philadelphia.* Lat. 39° 56′ 59′′ log. cos 9.88457 p cos q' log. 9.88517 p sin o' 2. Required the logarithms of p cos p' and p sin Ans. Log. p cos p Log. p sin o' log. sin 9.80762 log. y 9.99770 log. 9.80532 ', for Boston. : 9.82623 PROBLEM XVI. To calculate an Eclipse of the Sun for a given place, using the tables of the sun and moon contained in this work. For quantities independent of the place. 1. Find by prob. XII., the approximate time of new moon, in mean time at Greenwich, and let T represent this time, taken to the nearest whole hour. Put The latitudes and longitudes of various places are given in table VI. † When the approximate time of new moon, if expressed in time at the given place, would be as much as three or four hours before or after noon, it is generally better to take, for T, the whole hour which is an hour earlier or later, than the nearest whole hour. 2. For the time T, find, by prob. VI., the values of L', 8', ɛ, A', D', and E, and also the sun's hourly motion. To the value of L' at the time T, add the sun's hourly motion, and the sum will be the value of L' at the time (T+1 hr.). Also, for the time T, find, by prob. X., the values of L, λ, x, and the moon's hourly motions in longitude and latitude; and then, by means of the hourly motions, find the values of L and 2, for the time (T+1 hr.). 3. Using the values of the quantities at the time T, to log. A, taken from table IV., add Ar. Co. log. cos D', and call the result log. C. Expressing (LL) and a in seconds, to log. C add log. of (LL'), and also, to log. C add log. of x, and the sums will be respectively the logarithms of two quantities a and b. To log. of a and also to log. of b, add log. tang &, taken from table III., and log. cos L', and the sums will be respectively the logarithms of two quantities c and d. Attending to the signs of the quantities, subtract d from a, and call the result p; and add b and c together, and call the result q. с To log. B, taken from table IV., add log. tang 8', from table V., and the sum will be the logarithm of a quantity. To l' add 2732, and call the sum 7. To the logarithm 9.4180 and from it, add and subtract log. sin D', and call the results log. D and log. E. To the logarithm 8.250 add log. cos D', and call the sum log. M. at With the same log. C and log. tang &, and the values of L, L', and 2, the time (T + 1 hr.), find, as above, the values of p and q, for this time. Subtract the value of p, at the time T, from its value at the time (T + 1 hr.), and call the remainder p'. Do the same with the values of q, calling the remainder q'. With p' and ', which are the hourly changes of the values of p and q, and which may be regarded as constant during the eclipse, find the values of p and q for the times (T-1 hr.), (T — 2 hrs.), &c., and for (T + 2 hrs.), &c., by subtracting for the former and adding for the latter, and arrange them in a small table, as in the following exam ! ple. From the values of p and q thus found for whole hours, their values for any intermediate time may be easily obtained. Multiply 15° by the interval in hours between the time T and noon, the interval being marked negative when the time T is in the forenoon, and to the product add E. The sum will be the hour angle at Greenwich, at the time T. Call this hour angle H'. For quantities dependent on the given place. : 5. Find, by the last problem, log.p cos p' and log. p sin p', and, increasing the index of each by 4, call the results log. U and log. V. Then, using the value of D' at the time T, to log. U add log. sin D', and call the sum log. G. To log. V add log. cos D', and the sum will be the logarithm of a quantity f. Add together log. V, log. sin D', and the logarithm 7.668, and the sum will be the logarithm of a quantity a. Subtract a from 7, found by Art. 3, and call the result h'. These quantities may be regarded as constant during the eclipse. 6. To H', the hour angle at Greenwich at the time T, add the longitude of the given place, expressed in arc and marked affirmative when east, but negative when west, and the sum will be the value of H at the time T. Its value at any other time T', may be found by adding (T' T). 15°, found either by multiplication or from table XII., to its value at the time T. To find the approximate time of greatest obscuration. 7. Taking for p, q, and H, their values at the time T, to log. U and log. G add, respectively, log. sin H and log. cos H, and the sums will be the logarithms of two quantities u and g. To log. of u, add log. D, and to log. of g add log. E, and the sums will be the logarithms of two quantities and u'. Subtract g from f, and the remainder will be a quantity v. 8. To log. of (q′ — v′) add Ar. Co. log. of (p′ u'), and the sum will be the log. cotangent of an affirmative arc N, less than 180°. To log cot N add log of (qv), and the sum will be the logarithm of a quantity c. Add together twice log. sin N, log. of (p — u + c), and Ar. Co. log. of (jo u'), and the sum will be the logarithm of an interval of time ť. Then will Tt be the approximate time of greatest obscuration, in mean time at Greenwich. M To find the true* time of greatest obscuration, and approximate times of beginning and end 9. Taking T to represent the approximate time of greatest obscuration or nearly so, find p, q, and H, for this time; and then (Art. 7) find u, v, u', and '. To log. of u', add log. M, and the sum will be the logarithm of a quantity b. Subtract b from h', and call the remainder h. Find N, as in the last article, and to log. cot N add log, of (p — u), and the sum will be the logarithm of a quantity d. Add together log. sin N, log. of (d + v -9), and Ar. Co. log. of h, and the sum will be the log. cosine of an affirmative arc F, less than 180°. Add together log. cos (NF), log. of h, and Ar. Co. log. of (pu), and the sum will be the logarithm of an interval t. Add together log. cos (N — F), log. of h, and Ar. Co. log. of (p′ u'), and the sum will be the logarithm of an interval ť. And add together log. of (p u) and Ar. Co. log. of (p′ - u'), and the sum will be the logarithm of an interval t. Then will T t' + } (t + ť) be the true time of greatest obscuration; T't will be the approximate time of beginning; and T′ — t' + t' will be the approximate time of end. glą A To find the quantity of the Eclipse. 10. Add together the constant log. 1.0792, log. of h, Ar. Co. log. of (h — 2732), and twice log. sin F, or twice log. cosF, according as F is less or greater than 90°, and the sum will be the number of digits eclipsed; on the north limb when (d + v − q) is negative, but on the south when it is affirmative. T To find the true times of beginning and end. 11. Taking now T to represent the approximate time of beginning or nearly so, proceed, as in Art. 9, to find t and t", omitting the computation of t. Then will Tt- be the true time of beginning, very * The expression, true time, is to be taken here and in the subsequent part of the rule, in a relative sense; as only implying that the time found has an accuracy corresponding with that of the tables, from which the places of the sun and moon have been obtained, and of the number of decimals used in the calculation. With reference to a more exact determination, with more accurate data, they are near approximate times. They may frequently be in error to the amount of two or three tenths of a minute; and sometimes, perhaps, to the amount of half a minute. |