184 Transactions,— Miscellaneous, 
Casz 2.— When both angles are depressions. 
Using the same notation as before, except that D 
and d represent the true angles of depression, and 
D — R, d — R the observed angles of depression ; 
then D + d+ F = 2 right angles, also C + F = 2 
right angles ^. D+d=C;andD—-R+d—-—-R 
=C—2R. ae is, the sum de both angles of de- 
pression — C — —C—fC-i£OC,andiiC 
x 1$ x 153°6 (or sum of observed cbe in ers multiplied y 
177:8) = distance in links between A and D. If the distance between 4 
and B is required in feet, then multiply by 117 instead of 177:8. 
The above results expressed in words give the following 
Practical Rule. 
Take the sum of the observed vertical angles when both are depressions ; 
or their difference when one is an elevation, and reduce this sum or differ- 
ence to seconds; multiply by 177:3, and the result will be the approximate 
distance between the two stations in links. NorE.—lf the distance be 
required in feet, then multiply by 117. 
Or the following general rule will apply to all cases :—Subtract 180° 
from the observed zenith distances, reduce the remainder to seconds, and 
multiply by 177-3, the result wil be the approximate distance between the 
two stations in links. 
In the preceding investigation, I have assumed the mean value of 1" on 
the carth’s surface = 101:4 feet, and I shall now show what is the greatest 
error that can be introduced in any case by this assumption. 
'The radius of curvature on the meridian varies with the latitude from a 
minimum at the Equator e m to a maximum at the Pole (s. et 
And the radius of curvatare of the Prime Vertical also varies with the 
latitude from a minimum at the Equator (= E.) to a maximum at the Pole 
E 
P 
Also, the radius of curvature in any latitude varies with the Azimuth 
from a minimum on the meridian to a maximum on the prime vertical. 
Still the limits of variation are so small, compared with the ordinary 
errors of observation, that in general praetice it is sufficient to assume 
101:4 feet as the mean value of 1” on the surface of the earth for New 
Zealand. 
The following are the precise values for latitudes 89° and 44°, taking 89° 
as the mean latitude of the North Island of New Zealand, and 44? as the 
mean latitude of the South Island, 
cM E E d M RE E 
