PROBLEM XIII.

Given the Latitude of a place, to find the logarithms of cos and ę sin q', in which q' is the geocentric latitude and g the radius of the earth, for the given place.

To the cosine and sine of the given latitude, add respectively, log. a and log. y, taken from table XVII, with the latitude as the argument, and the sums will be log. e cos p ́and log. ę 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).

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EXAM. 1. Required the logarithms of e cos q' and ę sin q', for Philadelphia.*

2. Required the logarithms of cos p' and e sin p', for Boston.

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. X, the approximate time of new moon, in mean time at Greenwich, and let T represent this time, taken to the nearest whole hour.t 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',

', &, 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. VIII, the values of L, λ, %, and the moon's hourly motions in longitude and latitude; and then by means of the hourly motions, find the values of L and λ, 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. cos D', and call the result log. C. Expressing (LL) and A, in seconds, to log. C, add log. of (L-L'), 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 tang. taken from table III, and 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.

To log. B, taken from table IV, add tang. ', from table V, and the sum will be the logarithm of a quantity l'. To l', add 2732, and call the sum l. To the logarithm 9.4180, and from it, add and subtract sin D', and call the results log D and log. E. To the logarithm 8.250, add cos D', and call the sum log. M.

With the same log. B and tang. ɛ, and the values of L, L' and λ, at 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 q', 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 hr.), &c., and for (T + 2 hr.), &c., by subtracting for the former and adding for the latter, and arrange them in a small table, as in the following example. 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. e cos p' and log e 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 sin D', and call the sum log. G. To log. V, add cos D', and the sum will be the logarithm of a quantity f. Add together log. V, 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 sin H and 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 v' 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 cotangent of an affirmative arc N, less than 180°. To cot. N, add log. of (qv), and the sum will be the logarithm of a quantity c. Add together twice sin N, log. of (pu + c), and Ar. Co. log. of (p'-u'), and the sum will be the logarithm of an interval of time t'. Then will T-t', be the approximate time of greatest obscuration, in mean time at Greenwich.

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 v'. 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 cot N, add log. of (p—u), and the sum will be the logarithm of a quantity d. Add together sin N, log. of (d + v−q), and

* 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 corre sponding 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 inore 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.

Ar. Co. log. of h, and the sum will be the cosine of an affirmative arc F,

less than 180°.

Add together cos (N +F), log. of h, and Ar. Co. log. of (p' — u'), and the sum will be the logarithm of an interval t. Add together 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 + t'), be the true time of greatest obscuration ; T'—t"+t, will be the approximate time of beginning; and T' — t"+t', will be the approximate time of end.

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 sinF or twice cos F, 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 − y) is negative, but on the south when it is affirmative.

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 T'+t-t", be the true time of beginning, very nearly. Then, taking 'T' to represent the approximate time of end, find t' and t", omitting the computation of t, and T+t't", will be the time of the end.

To find an arc v, expressing the angular distance from the sun's vertex, of the point at which the eclipse begins or ends.

=

',

12. With the values of u and at the time T', for beginning, and their hourly changes of value u' and at that time, find the value of u and v, at the true time of beginning. Then using these values, to log. of u, add Ar. Co. log. of v, and the sum will be the tangent of an arc Q, less than 180°, which must have the same sign as u, and which will be numerically less or greater than 90°, according as v is affirmative or negative. Then V 270° + Q—(N + F) will be the distance of the point of beginning from the sun's vertex, reckoned to the right or west if V is affirmative, but in a contrary direction if V is negative. Finding in like manner, u and v, and then Q, for the true time of end, we have V = 270° + Q− (N −F). If u, v, and Q, be found for the true time of greatest obscuration, and we take V = 270° QN, when (d+v — q) at the approximate, time of greatest obscuration is affirmative, but V = 90QN, when (d+vq) is negative, then will V express, for the time of greatest

obscuration, the angular distance of the moon's centre from the sun's vertex reckoned as before, to the right or west.

ANNULAR OR TOTAL ECLIPSE.

13. If the value of (d+v-q), at the approximate time of greatest ob scuration, is numerically less than (h — 5464), or if it is so little greater, that the sum of cos N and log. of (d+v-q,) is numerically less than log. of (h5464), the eclipse will be total or annular; total, when (h-5464) is negative, but annular, when it is affirmative.

14. When it is ascertained that the eclipse will be total or annular, take N, F, (p' — u'), and t" as found for the approximate time of greatest obscuration (art. 9), and find t and t', using (h-5464), instead of h. Then will T' + t—t" and T' + t'—t", be the times at which the eclipse begins and ceases to be total or annular.

Note. The times time at Greenwich. place, by prob. V.

obtained by the above rules are expressed in mean They may be changed to mean time at the given

2. The above rule follows from the formulæ for eclipses, investigated in the appendix to part 1.

EXAMPLE.

It is required to calculate for Philadelphia, the eclipse of the sun of May 15th, 1836.

The approx. time of new moon is 15 d. 2 h. 8 m., Greenwich mean time.