L' = new latitude, computed from the formula for — dL.

And the value of dм in seconds of arc is obtained by converting K into seconds, by dividing K in metres by N sin 1', N being the normal, and the length of the radius used at that part of the earth in metres. The above formula thus becomes the one

already given (G). Lee's tables and formulæ gives a table for log. N, log.

1

N sin 1"

and log. (1 + e cos2 1.) for any latitude between 20 and 50 degrees. (G) is founded on the supposition that cos L' sin z к dм, whereas, in

450,

'3

(2 sin3 z — sin z). This is a maximum when z=24° 06',

and if we substitute this z in the latter expression, and make it equal to 0.001", we find the corresponding log. K to be 4.4315 =log. of 27000. For any line over 27000 metres, then a correction ought to be applied to dм, or if we will allow an error of 0.002, for any line over 34000 metres about 21 miles.

In the annexed table, the column headed du contains the log. of the seconds in a given arc; the column headed diff. contains the diff. between the log. of that arc and the log. of its sine (to the seventh place); the column headed к containing the log. of the length of that are in metres. To apply the correction in question after having first computed dм by the formula (G), enter the table with the given log. K, and take out the corresponding diff.; again enter the table with the computed log. dm, and take out the corresponding diff., and lastly, subtract the difference between the two quantities thus obtained from log. dm, the result will be the corrected log. dм.1

1 For denoting the difference between the log. arc and log. sin by 8, the formula (a) should be after the application of logarithms,

Log. A' should be carried to eight places. Seven places of logarithms should be used for dм.

For any line less than 340000 metres (21 miles) cos dɩ may be omitted, being regarded as 1.

In computing dz, sin [λ = † (L+1')] is taken out to five places for main chain of triangles, and to four for the others, carrying forward dz in tenths of seconds in the first, and in whole seconds in the second.

*The formula for the difference of azimuth is deduced as follows:-In the triangle rss' we have, by Napier's analogies, calling π, ' the polar distance of s, s',

But s' 1800-z', and sz, and cot (s' + s) = tan [900— (s' + s)] .'. n becomes

which is the formula (D) above, if we write dz, for tan (z'—z) and dм for tandм.

The formula for dz requires some amendment within the same limits, within which we obtain di and dм; we have

By transformation of (1) we get (see note p. 94, and make cos3 and cos3 of 1⁄2 dl =1)

We write the second term thus, tables, into which it can every half degree of L. 0.0002. The term dm3

dm3 F where log. F is to be taken from the easily be inserted, as only one value will be required for It is 7.8324 for 250, and 7-8404 for 450; diff. for 30' = can never exceed 0·1.

Whenever the log. of any term is not over 7.00.. the corresponding number need not be taken out.

Azimuths are reckoned from south round by west, and from 0° to 360°, the signs of sin z and cos z varying accordingly.

The following form, filled up with an example, is that at present used on the Coast Survey of the United States for the computation of the above formulæ for difference of latitude, longitude, and azimuth.

In this form the first horizontal line expresses the azimuth of the line joining the two stations A and B ; the second the angle formed by a line from A to a third station c, with AB; the third is found by the addition of these, and is the azimuth of Ac; the fourth the excess of the difference of azimuth between AC and CA over 180°, computed below at the bottom of the form; the fifth the azimuth of ca required, formed by the addition of the two above, and 180°. The sixth horizontal line contains the latitude L and longitude м of ▲; the seventh the difference of latitude and longitude of a

PROJECTION OF MAPS.

The geographical positions of the vertices of the triangles having been determined by calculation, as above explained, blank maps are prepared with lines upon them representing meridians and parallels of latitude, upon which these points are accurately put down in their true positions, and the maps thus prepared being placed in the hands of the plane table parties, are filled up with the details of the ground which they represent, in the manner described at p. 235 et seq., the points marked upon them, and identified upon the ground by the sunk masonry or pottery of the signals employed in the triangulation becoming the base points or points of departure for the operations of the plane table.

The mode of preparing these maps in practice, it will be now proper to explain.

A spherical or spheroidal surface like that of the earth not being developable, it is impossible to represent upon a plane the positions of places without changing more or less their distances from one another.* When a small portion of the earth's surface is to be represented, the best mode is to conceive the earth to be enveloped by a tangent cone, the

and c, computed below; the eighth the latitude L' and longitude м' of c, found by taking the algebraic sum of the two above. The next four horizontal lines of the form contain the computation of the difference of latitude dr between A and c, the first column being the computation of the logarithm of the first term, the second that of the second term, and so on, of the formula dL, at p. 325. The next two lines contain the four terms themselves,' and the next two their combination to form -dL. The first two columns of the remainder of the form contain the computation of dz; the third column that of the difference of longitude of A and c, viz. dм, and the fourth the correction of this, which is sometimes employed. Applied here the log. dm becomes 2.6873039, and dм 486.748, differing only 0.012 from what it was without the correction. A correction is also sometimes applied to dz, as has been already stated at p. 332, the formula for which is Fdm3, in which F= =12 sin A cos? A sin 1". The computation of this correction in the present example would be as follows:

In which F may be taken from a table previously prepared. The last number is the logarithm of the correction to be applied to dz.

* For the ordinary modes of projecting the hemisphere, see " spherical projections" in Davies' Descriptive Geometry, and for the analytical investigations of the same, see Francœur (Geodésie, 309, et soq.)

Iz being between 90° and 270°, cos z is negative, and ... h is negative.

circle of contact being the middle parallel of the region to be embraced, and to suppose the surface of this cone to coincide with that of the earth over the whole extent between the northern and southern parallels of the map. This cone, when developed, becomes the sector of a circle, the portion of which between the two extreme parallels which we have supposed to embrace the surface of contact, will represent the surface of the map.

Supposing the earth to be spherical, which may always be done in the projection of maps, its oblateness being so small, and representing the latitude of the middle parallel of the map by λ, and the number of degrees of longitude to be contained in the map by D, it is evident that the absolute length of the middle parallel of the map will be expressed by (see 1st note, p. 153)

In the above expression the radius of the earth is unity, and this being the case, the slant height or length of the element of the cone from the vertex to the circle of contact will evidently be the cotangent of the latitude. The arc of the sector, which is expressed by (1), divided by its radius cot λ, gives the length of the arc which measures the angle of the sector to radius unity. The result is

and this, which is the absolute length of the arc, must be multiplied by

180°

*to have its value, or the measure of the angle of the sector in

degrees,

D sin λ

(2)

then will be the formula for this angle, and the construction of the map will be very simple. It will only be necessary to draw two lines forming the angle expressed by (2), and with a radius equal to cos λ and the vertex of the angle as a centre, the arc representing the mean parallel may rd 180°

be described. If the map is to contain d degrees of latitude, then

will express the distance between the extreme parallels, and by describing arcs from the vertex of the sector with radii greater and less than

1800

=59°, 29573=3437', 74677=206264", 80625.