ApAus.— To Calculate Distances by Reciprocal Vertical Angles. 185 
FEET. 
Taking Bessel's value of the equatorial radius (E) = 20923597 
d sel’s value of the polar semi-axis ( == 20853654 
e value of 1” on the meridi la = 101:164 
The value of 1’ on the prime vertical = 101:575 
he mean value of 1” at all azimuths in lat. 39° — 101:370 
Again, the value of 1” on the meridian at | = 101-252 
A he value of 1” on the prime vertical at lat. 44° = 101-604 
é e alue of 1” at azimuths at - 449 = 101:428 
And the mean value of 1” at all azimuths at 39° E 101:370 
mean value of A atall azimuths for Wa Nude o[N A. = 1:999 
Or say, 101:4 fee 
It will thas be seen es by using this : mean value, the results would be 
sometimes slightly in excess of the true values, and sometimes slightly in 
defect; but in any case the difference would only amount to about 1 per 
cent., d may therefore in ordinary practice be neglected. 
With regard to the co-efficient of refraction which I have adopted, it 
may be thought that 45 is too small, as in most works on surveying it is 
stated to be from 4 to yy. 
The reason I have used 4; is because I find it more in accordance with 
actual observations in hilly country in New Zealand. 
The factor 177:3, as stated above, is obtained by taking the value of 1" 
on the carth’s surface as 153-6 links, and the refraction as 4 of the con- 
tained are; but if it is required to obtain the distance in any other 
denomination, such as feet, metres, miles, etc., for any other values of ter- 
restriul curvature and refraction, this may easily be done by means of the 
following formula :— 
Let v — value of 1" on the earth's surface, in the given denomination 
>> m-co-efficient of refraction 
>> £=the factor required ;~ then 
Foix 
Example. Suppose v— 80:89 metres and m=-071 
then — = = ae = 36, the factor required. 
It must be borne in mind ihat this method is only approximate, as the 
observed vertical angles are liable to an error of 2" or 3” even when an 
8-inch theodolite is used, and a mean of several observations taken. 
Supposing the average error of each double observation to be 5” or 6” 
then the error in the calculated distance would be 5 or 6 times 177 links, 
say about 10 chains. This would be 1 per cent. in a distance of 1000 
chains, which is the usual distance between geodesical stations in New 
Zealand. 
The chief ani of this method is that the observations are not 
subject to a ratio of error in proportion to the distance. Most approximate 
methods, by telemeters, etc., although tolerably correct for short distances, 
fail altogether when applied to long distances; but this method gives pro- 
