TRIGONOMETRICAL SURVEY OF NEW SOUTH WALES. 205 
in the number of observations. There is little doubt that the 
principal source of difficulty in trigonometrical levelling lies in the 
considerable and irregular changes in the atmospheric refraction. 
This cannot be better illustrated than by the experience obtained 
in India, where apparent zenith distances of points only 10 or 12 
miles away varied under certain circumstances through a range of 
as much as nine minutes. At first sight it might appear to be 
useless to go on duplicating such apparently wild observations, 
yet consideration of the nature of the question will show that the 
great range among the observations forms, itself, the reason for 
extending them to cover a longer period of time. Bessel’s opinion 
on this subject may be gathered from his estimate of the relative 
weights* to be assigned to such observations, the principal factor 
in the weight being n, n,, where n, and n, are the number of days 
on which observations are made at opposite ends of the same line. 
The differences of height are computed from the very con- 
‘venient formula— 
aes ater Ua (Ie oes ey 
1 
2 OU i 
’ where hand h, are the heights respectively of the initial station and 
that station the height of which is sought; € and @, are the reciprocal 
zenith distances observed at each station ; s represents the hori- 
zontal distance between the stations, and FR is the mean radius of 
curvature of the surface in the vertical plane passing through the 
two points. This formula involves only one trigonometrical 
function, and the values of A + h, and s’* are readily obtained 
2k 12 
from tables prepared with arguments h + h, and log s respec- 
tively. Considering the small number of observations referred to 
above, the deduced heights agree remarkably well both amongst 
themselves and in comparison with heights obtained by spirit- 
levelling, principally the levelling made in connection with the 
railway surveys. The only difference of any magnitude which has 
been revealed has been near the northern limit of the survey, 
where the heights of station Skeletar, close to Muswellbrook, 
deduced trigonometrically and through the spirit-levelling, differ 
by 9 feet. An adjustment to the railway levels has not in this 
case been made, as sufficient confidence was not felt in the accuracy 
of the railway levels. These were made some thirty years back, 
not under Governmental control, but by a contractor, and the 
writer is informed that where examined they have not been found 
* Grademessung in Ostpreussen, p. 196. 
