638

UNITED STATES COAST AND GEODETIC SURYEY.

The intersections Rx and B2 of the tangent pairs ax A, b3 B and a2 A, ô4 B (belonging to the ellipses Ex and E%, respectively) are situated upon a line Qx, forming one side of the polar triangle QxT, common to hoth ellipses. This line Qx intersects the diagonal AD in x and the quadrilatéral side BD in Q, and the lines drawn through Q from A and through x from B will intersect each other in the fourth point of intersection (S) of the two ellipses.

The quadrilatéral ABDS, obtained by connecting the four points of intersection of the two ellipses, has the point x as the intersection of its diagonals. By prolonging the sides BD and to their point of intersection Q and the sides AB and 8D to their point of intersection T, the three diagonal points QxT will form the polar triangle common to the two ellipses.

Also this method remains complicated and requires many lines to be drawn before the picture trace and the caméra station ( 8) may be plotted.

(3) Spécial application of the fixe-point problem for the case when the five points range themselves into a triangle.—The application of the “five-point problem” becomes very much simplified, how-ever, for the spécial case when the five points range themselves into a triangle, of which two sides (AG and CE) contain three points each (fig. 12).

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If we now place the strip of paper upon the radiais drawn from A, so that

e' falls upon AE, d1 falls upon AD, & falls upon A C,

it will hâve the position a2 b2 c2 d2 62, and the first ellipse (Ex) will résolve itself into the lines CE and Aa2.

If we now place the paper strip a! b' c' d'e' upon the radiais drawn from E to A, B and C, so that a' falls upon EA, b1 upon EB, and c' upon EC, it will assume the position ax bx cx dx ex, and the second ellipse (E2) will hâve resolved itself into the lines AC and Eex.

The intersection 8 of the two lines Aa2 and Eex will locate the station point with reference to the five given points, and by placing the paper strip upon the radiais SA, SB, SC, SD, and SE in such a way that a' falls upon SA, b’ upon SB, etc., its edge will locate the pfcture trace.

(4) To find the élévation (x) of a caméra station (S) that has been located by means of the “ five• point problemP—In order to ascertain the élévation of the unknown station S, platted after one of the preceding methods, it will become necessary to know the élévations of at least two of the five points.

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