Avams.—To Calculate Distances by Reciprocal Vertical Angles, 188 
as itis then only necessary to multiply the number of units in the given 
angle by the value of one unit at the earth's surface, in order to find the 
distance, or ** Contained Aro," as it is generally called. 
The method of deducing the subtended angle, or rather the length of the 
** Contained Are," when the vertical angle from each station to the other is 
given, may be shown as follows, dividing the problem iuto two eases—first, 
when one angle is an elevation ; and, secondly, when both are depressions. 
Case 1.— When one angle is an elevation, and the other a depression. 
i To investigate a method of ascertaining distances by 
means of reciprocal vertical angles:—Let 4 and B in 
EE fig. 1 represent the two stations, and C the centre of 
| the earth. Draw A F a horizontal line at A, and B G 
E a horizontal line at B, and B H parallel to A F. 
Then ÆA B G is the true angle of depression at B, and 
B A F' is the true angle of elevation at 4, and G B A 
— H B A (or BAF) = C (the contained are). Thus 
s we see that the difference between the true angles of 
elevation and depression is equal to the “contained 
arc," xcd d taking the mean value of l'on the earth's surface — 101:4 feet 
or 153-6 links, we could thus obtain the distance between the two stations 
in feet or links. 
But the observed angles are not the true angles, as they are both affected 
with refraction. 
Let 4 K and BK — the apparent direction of each station from 
the other. 
Let D pe. the true angle of depression, G B d. 
Let E represent the true angle of elevation B 4 F. 
Let C represent the angle 4 C B or contained are; and 
Let R represent the angle of refraction = K A B or K B A. 
G B K will be the apparent angle of depression = D — A. 
And KA F will be the apparent angle of elevation = E + R, 
and the difference between the observed angles of depression and elevation 
will be (D — R) — (E + R) = D — E — 2R = C — 2R. 
Now, assuming R to be 4, of the contained are, C — 2R will be C — $5 C 
asd» 
Therefore the difference between the observed angles of elevation and 
depression will be 13 C; so, by multiplying the number of seconds in 15 C 
by 13 x 158:6, we shall get the number of links in the contained are, or 
the distance between 4 and B. (Note 44 x 159:6 = 177:3). If the dis- 
tance between 4 and B is required in feet instead of links, then multiply 
by 1$ x 1014 = 117. 
