C—la

74

This has its least value when cot 2 A + cot 2 A is a minimum, Putting— y = cot 2 A + cot 2 2A ' X = 2 cot A + cot 2A sec 2 A = 0 dk ' :. cos 2 2A + 3 cos 2A =- 1 COS 2A rr v/5 ~ 8 2 A = 66" 15' nearly. The theoretical best shape is therefore an isosceles triangle, with the angles adjacent to the base, equal to 56" 15', and the apical angle to 67" 30. In such a triangle the unmeasured sides are less than the base in length, consequently a triangulation scheme could not be advantageously laid out with triangles of this form, but they can be used with advantage to expand a short base by the quadrilateral method. A point of great importance in a triangulation scheme is that the stations should be spread over the country at fairly regular intervals, and this is attained by triangles not differing much from the, equilateral form ; and for economical reasons the shape of the triangles is governed by the topography of the country to be surveyed, since the stations must usually be fixed on elevated points that command a clear view of the surrounding country. By plotting the curve y = cot 2 A + cot 2 2A for values of C between 0" and 180°, at intervals of 10°, a graphical representation is obtained, from which it is easy to see how far the angles of an isosceles triangle may safely depart from the equilateral form.

Ihe diagram shows that for isosceles triangles the angle at the apex may vary from 50" to 90" and the triangle remain well-conditioned. The ordinate increases rapidly for angles less than 30 or greater than 120" so that no angle in a triangle should lie outside these limits a well-known rule which is generally observed. The survey districts in the Dominion of New Zealand are about 124 miles square and the sides of the minor triangles are usually from three to five miles in length ; thus there are usually about twenty triangles in a district, generally depending on a measured base. The error in the summation of the three angles in a triangle, according to the latest regulations, is not to exceed .0 By taking 21 as the error m the summation of the triangles a correction of 7" for each angle may be used, and the error in an average set of triangles computed Taking the formula— f ±V j ,/ 2 (cot' 2 A I- cot 2 C + (§V [ ( 3 ) where C is the base-line and I in the error in the base-measurement, by following a chain of triangles winch leads to a check base or returns to the original base a comparison can be obtained of the measured and computed lengths, and the fractional error thus found can be used as a test of the accuracy of the angular results. ' The fractional error in the second triangle, using the computed value of (a) as a new base gives f' = ±p/!v 2 (cot 2 D + cot 2 F) + (4) =± -/ {-•' (cot 2 A + cof 2 C + cot 2 D + cot 2 F) + (-)*} by (1).