Plane trigonometry, by S.L. Loney.

ERRORS OF OBSERVATION.

465

EXAMPLES. LXVI.
1. The height of a hill is found by measuring the angles of elevation
a and p 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 0 in the
measurement of the angle a is 0. cos P sec a cosec (a - /) 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 8 in the observed value of a.
If a=1000 feet, a=30~, 3=15~, and there be an error of 1' 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
4 and 3 respectively. Find the height of the pole having given that
3 4
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 / 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
bec sin a sin feet.
sin (a - 3)
If AB be measured not directly from the tower but horizontally and
in a direction inclined at a small angle 0 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/ 02
quantity cos a sin - 2 must be subtracted.
cos / sin (a - /3) 2
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