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Table of Refractions.

Barom. 29.92 inc. Fah. Thermom. 54°.

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PROBLEM

PROBLEM XI.

To find the Angle made by a Given Line with the

Meridian.

1. The easiest method of finding the angular distance of a given line from the meridian, is to measure the greatest and the least angular distance of the vertical plane in which is the star marked a in Ursa minor (commonly called the pole star), from the said line: for half the sum of these two measures will manifestly be the angle required.

2. Another method is to observe when the sun is on the given line; to measure the altitude of his centre at that time, and correct it for refraction and parallax. Then, in the sphe

rical triangle zps, where z is the zerith of the place of observation, p the elevated pole, and s the centre of the sun, there are supposed given zs the zenith distance, or co-altitude of the sun, es the co-declination of that lu

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minary, Pz the co-latitude of the place of observation, and ZPS the hour angle, measured at the rate of 15° to an hour, to find the angle szp between the meridian pz and the vertical zs, on which the sun is at the given time. And here, as three sides and one angle are known, the required angle is readily found, by saying, as sine zs: sine ZFS :: sine ps: sine pzs; that is, as the cosine of the sun's altitude, is to the sine of the hour angle from noon; so is the cosine of the sun's declination, to the sine of the angle made by the given vertical and the meridian.

Note. Many other methods are given in books of Astronomy; but the above are sufficient for our present purpose. The first is independent of the latitude of the place; the second requires it.

PROBLEM XII.

To find the Latitude of a Place.

The latitude of a place may be found by observing the greatest and least altitude of a circumpolar star, and then applying to each the correction for refraction; so shall half the sum of the altitudes, thus corrected, be the altitude of, the pole, or the latitude.

For

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This method is obviously independent of the declination of the star: it is therefore most commonly adopted in trigonometrical surveys, in which the telescopes employed are

H

e

N

R

of such power as to enable the observer to see stars in the day-time: the pole-star being here also made use of.

Numerous other methods of solving this problem likewise are given in books of Astronomy; but they need not be detailed here.

Corol. If the mean altitude of a circumpolar star be thus measured, at the two extremities of any arc of a meridian, the difference of the altitudes will be the measure of that arc: and if it be a small arc, one for example not exceeding a degree of the terrestrial meridian, since such small arcs differ extremely little from arcs of the circle of curvature at their middle points, we may, by a simple proportion, infer the length of a degree whose middle point is the middle of that

arc.

Scholium.

Though it is not consistent with the purpose of this chapter to enter largely into the doctrine of astronomical spherical problems; yet it may be here added, for the sake of the young student, that if a = right ascension, d = declination, l = altitude, x = longitude, p = angle of position (or, the angle at a heavenly body formed by two great circles, one passing through the pole of the equator and the other through the pole of the ecliptic), i = inclination or obliquity of the ecliptic, then the following equations, most of which are newy obtain generally, for all the stars and heavenly bodies. 1. tan a = tan. cos i tan 1. sec. sin i. 2. sin d = sin λ cos l. sin i + sin l. cos i. 3. tan x = sin i . tan d. sec a + tana. cos i. 4. sin 1 = sin d. cos i sina.cos d. sin i.

5. cotan p = cos d.seca.cot i + sin d.tan a.

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10. tan * = tan A. cos i.) when 1 = 0, as is always the

11. cos λ = cos a cos d.

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case with the sun.

The

The investigation of these equations, which is omitted for the sake of brevity, depends on the resolution of the spherical triangle whose angles are the poles of the ecliptic and equator, and the given star, or luminary.

PROBLEM XIII.

To determine the Ratio of the Earth's Axes, and their Actual Magnitude, from the Measure of a Degree or Smaller Portion of a Meridian in Two Given Latitudes; the earth being supposed a spheroid generated by the rotation of an

ellipse upon its minor axis.

A

Dm

XM

CS

PS 00

E

P

Let ADBE represent a meridian of the earth, De its minor axis, AB a diameter of the equator, M, m, arcs of the same number of degrees, or the same parts of a degree, of which the lengths are measured, and which are so small, compared with the magnitude of the earth, that they may be considered as coinciding with arcs of the osculatory circles at their respective middle points; let мо, mo, the radii of curvature of those middle points, be = R and r respectively; MP, mp, ordinates perpendicular to AB: suppose further CD=c; CB = d; d2- c2 = e2; CP = x; cp = u; the radius or sine total = 1; the known angle вам, or the latitude of the middle point M, = L; the known angle Bsm, or the latitude of the point m, = 1; the measured lengths of the arcs M and m being denoted by those letters respectively.

Now the similar sectors whose arcs are M, m, and radii of curvature R, r, giveR:r::M:m; and consequently RM= The central equation to the ellipse investigated at p. 29 of this volume gives PM = √(d-x2); pm = √(d2-u2);

Μ.

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also SP = ; sp = (by th. 17 Ellipse). And the method of finding the radius of curvature (Flux. art. 74, 75), applied to the central equations above, gives

R=

(d4-e2x2)

c4d

; andr =

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c4d

On the other hand, the triangle SPM gives SP: PM :: cos L : sin L'; that is, (d2-x2):: cos L: sin L; whence r2 =

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d+ cos2 L da-e sin L d4 cos l

And from a like process there results, u2 =

d2e2 sin2 l

Substituting in the equation Rm = rm, for R, and r their values, values, for r2 and u2 their values just found, and observing that sin2 L + cos2 L = 1, and sin2l + cos2 1 = 1, we shall find

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Whence, by extracting the root, there results finally

d

c

(M3 sin L + m3 sin 1). (Msin L
(m3 cos l + M COS L). (m3 cos l

M sin l)

M COS L)

I

M3 COS L)

This expression, which is simple and symmetrical, has been obtained without any developement into series, without any omission of terms on the supposition that they are indefinitely small, or any possible deviation from correctness, except what may arise from the want of coincidence of the circles of curvature at the middle points of the arcs measured, with the arcs themselves; and this source of error may be diminished at pleasure, by diminishing the magnitude of the arcs measured : though it must be acknowledged that such a procedure may give rise to errors in the practice, which may more than counterbalance the small one to which we have just adverted.

Cor. Knowing the number of degrees, or the parts of degrees, in the measured arcs M, m, and their lengths, which are here regarded as the lengths of arcs to the circles which have R, ", for radii, those radii evidently become known in magnitude. At the same time there are given the algebraic values of R and r: thus, taking R for example, and extermi

nating e' and r2, there results R =

ds

.There

c(d2- (d2-c2) sin2 L)2 fore, by putting in this equation the known ratio of d to c, there will remain only one unknown quantity d or c, which may of course be easily determined by the reduction of the last equation; and thus all the dimensions of the terrestrial spheroid will become known.

General

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