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Dividing therefore N into equal parts, and for each point of division finding the corresponding value of x from the above formula, so many points in the arc NN' will be determined.

p is known since pr -8. It remains only to find in order to have for each point like м the values of the co-ordinates x and y, viz:

x=p sin 0, y=8+x tan } 0

N. B. That s must be taken negative when λ < 1.

The longitude of м estimated from the central meridian suppose to be A. This will also be the number of degrees in the arc of the parallel.

But N being the normal at the point м, terminating at the polar axis, the radius of this parallel will be (see diagram, p. 365, App. VI.),


Moreover, the arc Bм of the projection is of the same length with the arc of the parallel, but the number of degrees in two arcs of the same length will be in the inverse ratio of their radii

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which formula serves to determine @ in the same denominations that A is given. N is known in terms of λ from formula

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(see App. VI., p. 368), in which e=0.0816967, log. e 8.9122052271.

It is easy to perceive now how a map would be drawn according to the projection under consideration. Two lines AC and AX are first drawn at right angles to each other, intersecting at the middle of the sheet A. Setting out from A, we lay off above and below distances such as AB, respectively equal to the values of s, that is to say to arcs of the meridian corresponding to 10, 20, 30, ... of distance from A, arcs which go on increasing towards the pole. Next we compute the values of the normals N, N', . . . from degree to degree, the radii p of the projected parallels, and finally the amplitudes of the angles 6, which correspond to values of A and A, varying also by degrees, whence result the co-ordinates x and y of the vertices of quadrilaterals in which meridians and parallels of latitude distant from each other, respectively the space of 10 intersect. It remains only to lay off these co-ordinates by a scale of equal parts. The sides of the quadrilaterals joining these vertices thus determined may be drawn without sensible error as straight lines.

The territory to be represented by the map is ordinarily too extended to be placed

1 For their values see p. 367, App. VI., also Lee's Tables and Formulas, p. 84, Part II.



The zenith sector employed on the coast survey of the United States for determining latitude astronomically, is the same as the mural circle already described, p. 306, except that only two small portions of the limb, the one above, the other below the centre, are retained, the rest being conceived to be cut away, to render the instrument more portable. The limb and telescope, instead of being sustained by a wall, are attached to a vertical flat beam of iron, which is capable of reversal about a vertical axis, and also end for end. Long spirit levels can be attached to the

upon a single sheet. It is customary to form the map by the union, border to border, of a series of sheets, the dimensions of which are 8 decimetres by 5 To find the positions of the vertices of the quadrilaterals upon these sheets, the origin of co-ordinates is transferred to one of the corners of the sheet, an operation which consists simply in adding or subtracting 1, 2, 3,... times 8 decimetres in the direction of the x, and as many times 5 decimetres in the direction of the y3, according to the place which the sheet ought to occupy in the assemblage. The order of the sheets is marked upon them. Thus the sheet 3 is the one which is second in the horizontal direction, and third in the vertical, estimating from a the intersection of the middle meridian and parallel.


As to the inverse problem, to find the latitude and longitude of a point given upon the map, it will be sufficient to draw through the point lines parallel to the sides of the quadrilateral within which it falls, and to determine upon the scale of equal parts, the values of the fractions which the lines represent.

The following formulas are used on the United States Coast Survey, when the extent of the map is not more than 40 of latitude and longitude.

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Here the cone,

instead of being assumed tangent to one of the parallels of the map, is supposed successively tangent to each, that it may be required to draw upon

back of this bar, in addition to the ghost apparatus in front*. Zenith telescopes are also employed, of similar but less elaborate construction. The practice on the coast survey with these instruments is to observe two stars near the zenith, one north of it, the other south, and differing so little in zenith distance that the difference may be measured with a micrometer.t

The following is a very full exposition of the theory of this method.

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it. The map thus becomes the developement of the surfaces of several successive


To make the projection, a central meridian is drawn upon the map, along which the lengths of the required minutes are laid off; perpendicular lines are drawn at each of these points, and the values of dp and 8m are laid off successively, along and from each of these lines.

Lee's Tables give the values of op and ôm for parallels 30' apart. The manuscript tables in use on the Coast Survey are computed to every minute. In the diagram, MB, which is very small, is regarded as a straight line, and the angle Q

cisz; CM is equal to MKT in the diagram on p. 365, App. VI., N = MN, and LMNV in that diagram. With these explanations the student will readily deduce the above formulas for dp and sm.

* The mode of observing for latitude is similar to that employed with the mural, except that the readings of three levels, one above the other, at the back of the bar, are taken, and the observation is repeated upon the same star in reversed positions of the instrument. The correction for the state of the levels and the reduction to the meridian are made on the principles indicated at pp. 343, 344.

+ For description of micrometer see p. 362, App. VI.

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Now if the observation of a star were not affected by refraction, and it were observed at the moment it passed the meridian, and the instrument at the same time were perfect as to level, then the latitude resulting from the observation of a star north of the zenith would be expressed by

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But, since every observation on a star is affected more or less by refraction, according to its distance from the zenith, and as the instrument is constantly changing (as indicated by the level) during the observations, it becomes necessary that these quantities should be known, and the observations corrected accordingly.


Suppose a star to be observed north of the zenith, and its north polar distance to be equal to NA in the diagram. If there were no refraction, the star would be observed at A, and the zenith distance would be Az. But by the effect of refraction, which elevates an object, the star is seen and observed at a, consequently the observed zenith distance of the star is equal to az, which is too small, since NA is the true N. P. D. of the star. The measured z. D. must therefore be increased by sa, or the amount of refraction.

Hence we have, as a result for latitude by observing a star north of the zenith,

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Having observed the star north of the zenith, the telescope is turned

in azimuth 1800, for the purpose of observing the star south of the zenith the N. P. D. of which is equal to NB.

On account of refraction, the star is observed at b, and the measured zenith distance is zb, which is evidently too small, and must be increased by вb, the amount of refraction.

For a star south of the zenith we have

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If a star north of the zenith be supposed to have been observed at a, while the instrument was perfectly level, but by a sudden change in the temperature of the atmosphere or any other cause, the instrument is thrown out of level, and the vertical axis takes the line sc; since the instrument was perfectly level when the observation was made, the north end of the level scale must have read the same as the south end (supposing the level to be in adjustment).

Now, the level being in the direction due north and south, and at right angles to the vertical axis of the telescope, it is evident that when the vertical axis inclines towards the south, the north end of the level will become elevated, and the south end depressed; and therefore the north end of the level scale will read greater than the south end.

Suppose another star of the same north polar distance as the one just observed, should come into the field of the telescope, it would be seen at a, and consequently the distance ac must be measured with the micrometer, thereby increasing the true z. D. of the star by the quantity ae = zc.

Since this distance zc is also measured by half the difference (in arc) of the readings of the level scale (see p. 155), it follows that the level correction will always equal the difference between the measured z. D., and the true z. D. of the star.

Hence we have for latitude, by observation of a star north of the zenith,

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The telescope is next turned in azimuth 180°, for the purpose of observing a star south of the zenith. The vertical axis still being in the line sc, the telescope will take the line sb, but the star will be seen at d, therefore the measured z. D. will be too small by the quantity bd = zc. The distance zc being measured (as I have before stated), by half the dif

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