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For, suppose the shorter radius equal to AL, and hence, IN equal to AI; join LN, then the equal triangles AIL and NIL give AL = LN; so that the curve, if continued, will pass through N, where its tangent will coincide with IN. Then (Art. 365) the common tangent point would be the intersection of the straight line through B and N with the first curve; but in this case there can be no intersection, and therefore no common tangent point.

Suppose, next, that this shorter radius is greater than AL, and continue the curve till its tangent becomes parallel to BI. In this case, the extremity of the curve will fall outside the tangent BI in the line AN produced, and a straight line through B and this extremity will again fail to intersect the curve already drawn. As no common tangent point can be found when the shorter radius is taken equal to AL, or greater than AL, no compound curve is possible. This radius must, therefore, be less than AL.

In like manner it may be shown that the radius of the other branch of the curve must be greater than BM.

If the tangents AI and BI, and the intersection angle I are known, then

AL AI tan I;

BM BI tan I.

These values are, therefore, the limits of the radii in one direction.

367. If nothing were given but the position of the tangents and the tangent points, it is evident that an indefinite number of different compound curves might connect the tangent points; for the shorter radius might be taken of any length less than the limit found above, and a corresponding value for the greater could be found. Some other condition must, therefore, be introduced, as, for example, in the following problem:

368. Given the line AB a (Fig. 177), which joins the fixed tangent points A and B, the angle BAI = A, the angle ABI = B, and the first radius AE= R, to find the second radius BF R' of a compound curve to unite the tangents HA and BK.

Let the first curve be run with the given radius from A to D, where its tangent DO becomes parallel to BI; then the common tangent point C is in the line BD produced, and the chord CB CD+BD.

The angle OAD, formed by a tangent and a chord, is measured by half the arc ACD; hence, R sin OAD = R sin IADAD (see Legendre, Trig. Arts. 64 and 30); hence,

AD2R sin IAD.

(1)

In the triangle OAD, since OA and OD are equal, angle AOD= 180°-20AD; hence, OAD=IAD=90°—†AOD=90°—†AIB; but from the triangle AIB, AIB = 90° —† (A+B);

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Then in the triangle ABD, we have AB = a, AD = 2R sin (A+B), and the included angle

DAB = IAB—IAD = A—† (A + B) = † (A−B) ;

whence, we have the proportion (Legendre, Trig. Art. 45),

AB+AD: AB-AD :: tan (ADB+ABD)

† (ADB+ABD)

=

: tan (ADB-ABD). (4)

† (180° — DAB); } (ADB—ABD) may be found from (4); and from the half-sum and half-difference thus

obtained, the angles ADB and ABD may be found. These

angles being known the side BD may be found from the

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Substituting these values of CB and CD, in the equation

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When the angle B is greater than A, that is, when the greater radius is given, the solution is the same except that the angle DAB (B-A), and CBI is found by subtracting the supplement of ABD from B. We shall also find CB-CD-BD; hence,

=

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NOTE.-If more convenient, the point D may be determined in the field by laying off the angle IAD (A + B), and measuring the distance AD = 2R sin † (A+B). BD and CBI may then be measured, instead of being calculated as above.

EXAMPLE. a 950, A = 8°, B=7°, R 3000; find R'.

=

: =

AD 2 x 3000 sin (8°+7°) = 783.16

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log tan (ADB-ABD), 87° 24' 17" 1.343641

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BOOK X.

MINING

SURVEYING.

SECTION I.

DEFINITIONS AND GENERAL PRINCIPLES.

369. Mining Surveying comprises all the operations necessary to determine the relative positions of the parts of a mine with respect to each other, and also with respect to the surface of the earth.

370. The general principles involved in this branch of surveying are the same as those used in surface surveying, but, from the nature of the case, certain modifications are required.

Stations are designated by lamps instead of flags, and lampstands instead of flag-rods, or by plumb-lines properly suspended and lighted; station points, if temporary, are marked by crosslines chipped in the rock, or sometimes by simple chalk lines, and, if permanent, by iron pegs driven into holes drilled for the purpose.

The compass is rarely used for underground work, and ought never to be used for any but rough work, because of the inaccuracies to which it may lead. A great deal of the mining in the U. S. is done in the far West, where the magnetic declination is not known; and, further, the declination is seriously affected by the state of the atmosphere, the presence of iron ores, magnetic pyrites, &c.

The transit, which is the principal angular instrument em

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