CHAPTER XXXI. ERRORS OF OBSERVATION. 391. WE have up to the present assumed that it is possible to observe any angles perfectly accurately. In practice this is by no means the case. Our observations are liable to two classes of errors, one due to the instruments themselves, which are hardly ever in perfect adjustment, and the other class due to mistakes on the part of the observer. 392. An error in any of our observations will clearly, in general, cause an error in the value of any quantity calculated from that observation. For example, if in Art. 192 there be a small error in the value of a, there will be a consequent error in the value of x which, as we see from the result of that article, depends on a. 393. The importance of an error in a length depends, in general, upon its ratio to that length. For example in measuring a piece of wood, about six feet long, a mistake of one inch would be a very serious error; in measuring a mile racecourse a mistake of one inch would be not worth considering; whilst in measuring the distance from the Earth to the Moon an error of one inch would be absolutely inappreciable. 394. We shall assume that the errors we have to consider are so small that their squares (when measured in radians if they be angles) may be neglected and we shall give some examples of finding the errors in derived quantities. We shall assume that our tables and calculations are correct, so that we have not to deal with mistakes in calculation but only with errors in the original observation. 395. Ex. 1. MP (Fig. Art. 42) is a vertical pole; at a point O distant a from its foot its angular elevation is found to be 0 and its height then calculated; if there be an error d in the observation of 0 find the consequent error in the height. The calculated height ha tan 0, clearly. If the error d be in excess, the real elevation is -8, and hence the real height h’= a tan (0 – 8). if we neglect squares and higher powers of d. The ratio of the error to the calculated height Except when sin 20 is small this ratio is small since & is small. It is The ratio is large when is near zero and when it is near 2 Hence a small mistake in the angle makes a relatively large mistake in the calculated result when the angle subtended is very small or when When is small, both the calculated height and the absolute error, viz. a tane and a sec20.d, are small, but the latter is great compared with the former. When is nearly 90°, both these quantities are great. Ex. 2. The height of a tower is found as in Art. 192; if there be an error 0 in excess in the angle a, find the corresponding correction to be made in the height. The real value of a is a-; hence the real value of the height is found by substituting a 0 for a in the obtained answer, and therefore The error in the calculated height is therefore ◊. one of excess. Also the ratio of the error to the calculated height 0 sin ẞ Ex. 3. The angles of a triangle are calculated from the sides a=2, b=3, and c=4, but it is found that the side c is overestimated by a small quantity &; find the consequent errors in the angles. From the given values of the sides we easily have Corresponding to the value 4-8, let the values of the angles be A – 01, so that the smallest angle has the least error. We note, as might have been assumed a priori, that the sum of the errors in the three angles is zero. This is necessarily so, since the sum of the angles of any triangle is always two right angles. EXAMPLES. LXVI. 1. The height of a hill is found by measuring the angles of elevation a and ẞ of the top and bottom of a tower of height b on the top of the hill. Prove that the error in the height h caused by an error ✪ in the measurement of the angle a is . cos ß sec a cosec (a – ẞ) times the calculated height of the hill. ་ 2. At a distance of 100 feet from the foot of a tower the elevation of its top is found to be 30°; find the greatest and least errors in its calculated height due to errors of 1' and 6 inches in the elevation and distance respectively. 3. In the example of Art. 196 find the errors in the calculated values of the flagstaff and tower due to an error d in the observed value of a. If a=1000 feet, a=30°, ß=15°, and there be an error of l′ in the value of a, calculate the numerical value of these errors. 4. AB is a vertical pole, and CD a horizontal line which when produced passes through B the foot of the pole. The tangents of the angles of elevation at C and D of the top of the pole are found to be 4 3 3 and respectively. Find the height of the pole having given that CD=35 feet. Prove that an error of 1' in the determination of the elevation at D will cause an error of approximately 1 inch in the calculated height of the pole. 5. The elevation of the summit of a tower is observed to be a at a station A and B at a station B, which is at a distance c from A in the direct horizontal line from the foot of the tower, and its height is thus found to c sin a sin ß sin (a - ẞ) be feet. If AB be measured not directly from the tower but horizontally and in a direction inclined at a small angle @ to the direct line shew that, to correct the height of the tower to the second order of small quantities, the c cos a sin2 6 02 cos ẞ sin (a - ẞ) quantity 2 must be subtracted. 6. A, B, and C are three given points on a straight line; D is another point whose distance from B is found by observing that the L. T. 30 |