B

263. Three-point Problem.-The object sought in the solution of this problem is the determi

Pp

nation of the unknown position of an occupied station P, when the positions y of three other stations, A, B, and C, are (See Graphic Solution, Art.

FIG. 173.-THREE-POINT
PROBLEM.

in which

known.

75.)

The problem is indeterminate when P is on the circumference of a circle passing through A, B, and C. This is known by the sum of the angles pc, being equal to 180°, and also by the radius of the circumference passing PAC, being equal to that for PBC.

cot x = cot R =

1. R = 360° - p- p' — c or R = x - y or R x = y.

2. log

If ppc or nearly, the solution is impossible.

a sin p'

b sin

cos R.

(61)

log of a number taking the sign of

3. Add algebraically +1 to the above number.

4. Take out the log of this number, annexing the proper

sign.

THREE-POINT PRoblem.

601

5. Then add this log to log cot R, remembering that this is in effect multiplying one by the other, and the rule of the signs must be attended to; this gives the log of the cot of x. Then Rx = y.

6. If R < 90°, cos R is + and cot R is
"R< 270° and R > 180°,
"R< 180° and R > 90°,

"R< 360° and R > 270°,

cot R is +.

cot R is

+.

cos R is

cos Ris

cos R is +;

cot R is —.

7. p' is at opposite side of quadrilateral to a,

and p to b.

8. The angle is always the interior angle of the quadrilateral PBCA, and C is the middle point as seen from P.

EXAMPLE. The following quantities are known from observation or computation, since the positions of B, C, and A are known, namely:

CHAPTER XXVIII.

ADJUSTMENT OF PRIMARY TRIANGULATION.

264. Method of Least Squares.-By the method of least squares is understood a process by which observations are adjusted and compared. When several precise measurements have been made of a given quantity, no matter how similar the conditions may appear, the results do not agree and it becomes necessary to adjust the various measures or observations in order to get a mean or apparent agreement. The result is not necessarily the true value, but is used and accepted as such since it is a mean derived from the combination and adjustment of all the measures taken which are most probably and apparently correct.

Errors of observation are of two kinds, (1) systematic and (2) accidental; the former, resulting from unknown causes, affect all observations alike, while accidental errors are of a kind which produce discrepancies between observations: and it is this kind of errors alone, and not the systematic errors, which are considered in the so-called "theory of errors" and which it is the object of adjustment to minimize. The error of observation is truly the difference between the observed and true value, and may be plus or minus according as it exceeds or is less than the true value. The object of the theory of errors is to obtain from a number of discordant observations the best obtainable result. The fundamental principle of the method of least squares is the rule of Legendre, that, in observations of equal precision the most probable values

METHOD OF LEAST SQUARES.

603.

of observed quantities are those that render the sum of the squares of the residual errors a mininum.

The probable value of an observed quantity is that which we are justified in considering as the more likely to be the true value than any other. As stated by Prof. Mansfield Merriman, the probability is expressed by an abstract fraction, which measures numerically the degree or likelihood in the happening or failing of an event; as confidence may range from improbability to certainty, so this measure may range from zero to one. If the figure 6, for example, occurs once on a die of six faces, the probability of its turning up when thrown is ; likewise, if the same figure occurs on each face, the probability of its turning up when the die is thrown is, or unity, which is certainty.

When a number of unknown quantities are to be determined by means of equations involving unknown quantities, the quantities sought are said to be indirectly observed. necessary to have as many such indirectly observed equations as there are unknown quantities, and the discovery of these unknown quantities by solution of equations is the method of least squares. The differences between the several observed values and that which is taken as the true value of an observation are called the residuals, and these are the apparent errors of observation. When observations are not made under the same conditions and the computer is aware of reasons which prevent them being equally good, a greater relative importance may be given to better observations by treating them as equivalent to more than one occurrence of the same value in a set of equal observations; in other words, they may be weighted. Weights (Art. 284) may therefore be regarded as numerical measures of the influence of the observation upon the arithmetical mean.

In observing a series of angles, the angles read at station Walton, between points n and o= a, o and pb, p and q = c, and q and r = d (Fig. 175, p. 613) are rendered functions of an adjustment equation. If combinations of these angles are

=

observed, as the angles between n and pg, and between o and q i, and between n and q = g+c, then the means of the various angles separately measured between n and o = a, and o and p = b, should be equal to the mean of the angle read between n and p = g. Similarly with the others, and by thus observing all of the separate and many of the combined angles it becomes possible to arrange a number of equations of condition, as they are called. In these, however, the mean observed angles never exactly sum up as they should theoretically, and the differences are called the residuals.

265. Rejection of Doubtful Observations. When theodolites or other angle-measuring instruments are used, there occur among a number of observations for the value of a particular angle, one or more which differ greatly from the mean of all. It is not advisable to depend entirely on judgment as to which of these observations shall be retained and which rejected. The least objectionable criterion by which to judge as to the rejection of doubtful observations, and one based on mathematical principles, has been stated by Mr. T. A. Wright thus:

Where an observation differs from the general run of the series by more than five times the probable error or three times the mean square error, attention should be called to it.

An excellent fixed rule for the rejection of doubtful observations is Peirce's Criterion, which is applied in the following

manner:

Let m number of measures;

n = number of doubtful observations to be rejected

e

(to be found by trial);

mean error of one observation in the set of m;

v, v', etc. = residuals of the observations or the difference of

each value from the mean; and

x = ratio of required limit of error for the rejection of n observations, to the mean error e; so that

xe is the limiting error.