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feet in the radius of the Earth. Then if E be the circular measure of the spherieal excess,
Now by actual measurement the mean length of a degree on the Earth's surface is found to be 365155 feet; thus
With the value of r obtained from this equation it is found by logarithmic calculation, that
log n = log 8 – 9.326774. Hence n is known when s is known.
This formula is called General Roy's rule, as it was used by him in the Trigonometrical survey of Great Britain and Ireland. Mr Davies, however, claims it for Mr Dalby. (See Hutton's Course of Mathematics, by Davies, Vol. 11. p. 47.)
122. In order to apply General Roy's rule, we must know the area of the spherical triangle. Now the area is not known exactly unless the elements of the spherical triangle are known exactly ; but it is found that in such cases as occur in practice an approximate value of the area is sufficient. Suppose, for example, that we use the area of the plane triangle considered in Legendre's Theorem, instead of the area of the Spherical Triangle itself ; then it appears from Art. 109, that the error is approximately
a® + B + y denoted by the fraction
of the former area, and this
24r fraction is less than •0001, if the sides do not exceed 100 miles in length. Or again, suppose we want to estimate the influence of errors in the angles on the calculation of the area; let the aß sin C circular measure of an error be h, so that instead of
2 aß sin (C+) we ought to use
; the error then bears to the area
2 approximately the ratio expressed by h cot C. Now in modern observations h will not exceed the circular measure of a few seconds, so that, if C be not very small, h cot C is practically insensible.
123. The following example was selected by Woodhouse from the triangles of the English survey, and has been adopted by other writers. The observed angles of a triangle being respectively 42°. 2. 32", 67o. 55'. 39", 70°. 1'. 48", the sum of the errors made in the observations is required, supposing the side opposite to the angle A to be 27404.2 feet. The area is calculated from the ex
a® sin B sin c pression
and by General Roy's rule it is found 2 sin A that n= •23. Now the sum of the observed angles is 180° — 1", and as it ought to have been 180° + •23", it follows that the sum of the errors of the observations is 1".23. This total error may be distributed among the observed angles in such proportion as the opinion of the observer may suggest; one way is to increase each of the observed angles by one-third of 1"-23, and take the angles thus corrected for the true angles.
124. An investigation has been made with respect to the form of a triangle, in which errors in the observations of the angles will exercise the least influence on the lengths of the sides, and although the reasoning is allowed to be vague it may be deserving of the attention of the student. Suppose the three angles of a triangle observed, and one side, as a, known, it is required to find the form of the triangle in order that the other sides may be least affected by errors in the observations. The spherical excess of the triangle may be supposed known with sufficient accuracy for practice, and if the sum of the observed angles does not exceed two right angles by the proper spherical excess, let these angles be altered by adding the same quantity to
each, so as to make their sum correct. Let A, B, C be the angles thus furnished by observation and altered if necessary; and let 8A, 8B and 8C denote the respective errors of A, B and C. Then 8A + B + 8C = 0, because by supposition the sum of A, B and C is correct. Considering the triangle as approximately plane, the
a sin (C+8C)
a sin (C + 8C) true value of the side c is
sin (A +84) sin (A - B -80) Now approximately
sin (C + 8C) = sin C + 8C cos C, (Plane Trig. Chap. XII.),
Now it is impossible to assign exactly the signs and magnitudes of the errors 8B and 8C, so that the reasoning must be vague.
It is obvious that to make the error small sin A must not be small. And as the sum of 8A, 8B and 8C is zero, two of them must have the same sign, and the third the opposite sign; we may therefore consider that it is more probable than any two as 8B and 8C have different signs, than that they have the same sign.
If 8B and 8C have different signs the errors of b and c will be less when cos A is positive than when cos A is negative ; A therefore ought to be less than a right angle. And if 8B and 8C are probably not very different, B and C should be nearly equal. These conditions will be satisfied by a triangle differing not much from an equilateral triangle.
If two angles only, A and B, be observed, we obtain the same expressions as before for the errors in b and c; but we have no reason for considering that 8B and 8C are of different signs rather than of the same sign. In this case then the supposition that A is a right angle will probably make the errors smallest.
125. The preceding article is taken from the Treatise on Trigonometry in the Encyclopædia Metropolitana. The least satisfactory part is that in which it is considered that SB and 8C may be supposed nearly equal; for since 8A + B + 8C = 0, suppose SB and 8C nearly equal and of opposite signs, we do in effect suppose 8A = 0 nearly; thus in observing three angles, we suppose that in one observation a certain error is made, in a second observation the same numerical error is made but with an opposite sign, and in the remaining observation no error is made.
126. We have hitherto proceeded on the supposition that the Earth is a sphere; it is however approximately a spheroid of small eccentricity. For the small corrections which must in consequence be introduced into the calculations we must refer to the works named in Art. 114. One of the results obtained is that the error caused by regarding the Earth as a sphere instead of a spheroid increases with the departure of the triangle from the well-conditioned or equilateral form (An Account of the Observations... page 243). Under certain circumstances the spherical excess is the same on a spheroid as on a sphere (Figure of the Earth in the Encyclopædia Metropolitana, pages 198 and 215).
127. In geodetical operations it is sometimes required to determine the horizontal angle between two points, which are at a small angular distance from the horizon, the angle which the objects subtend being known, and also the angles of elevation or depression.
Suppose, OA and OB the directions in which the two points are seen from 0; and let the angle AOB be observed. Let OZ be the direction at right angles to the observer's horizon ; describe a sphere round 0 as a centre, and let vertical planes through 04 and OB meet the horizon at OC and OD respectively : then the angle COD is required.
Let AOB=0, COD= 8 + 2, AOC =h, BOD=k; from the triangle A ZB
This formula is exact; by approximation we obtain
cos 0 - hk cos 0 - 2 sin e
1-3(h* + k*) )