TRIGONOMETRICAL SURVEY OF NEW SOUTH WALES. 225 
RELATION BETWEEN ASSUMED AND ACTUAL SURFACES. 
It may be shown that the effect of projecting the points of a 
triangulation upon a spheroid of reference which must, by the 
method of the problem, differ only slightly from the actual 
spheroid, will produce differences which are only of the second 
order in comparison with the inclination of the surfaces to each 
other, and further, that these differences are so small as to be 
overwhelmed by other sources of error. Supposing, therefore, 
that € and 7 are the inclinations between the surfaces at one 
point, € in the direction of the meridian, and 7 in a plane at right 
angles thereto, we are enabled by a comparison of the astronomical 
and geodetic positions to express the € and 7’ belonging to any 
other point in terms of € and y, and, by determining a set of 
elements for the earth’s figure, such that the sum of the squares 
of all the £s and 7’s is a minimum, we arrive at the most probable 
shape of the surface. From this treatment there result for the 
latitude, longitude, and azimuth observations, equations of the 
following forms :— 
f@=xn +AE+By+ Cu+Etnv 
sec fy =x + A'E+ By + Cut Ev 
tan f'n = «6 + A”E + Bn + Cu + E’v 
in which zw is a correction on the assumed value of the major semi- 
axis, and v a correction on the eccentricity of the elliptic meridian. 
DIMENSIONS OF MOST PROBABLE SPHEROID. 
Sixty-eight equations of the above form have been obtained, 
from which, allowing the same weight to the longitude and azimuth 
equations as to the latitude equations, normal equations are got 
by the method of least squares, the solution of which gives the 
following values :— 
€ = — 0:45"; » = — 1:39"; w = + 0°550; v= + 1518. 
With these values the elements of the spheroid would appear 
to be :— 
a = 6974378 + 1861 yards; e? = :0073875. 
From an examination of the data we would be led to expect 
that certain systematic influences would to some extent affect this 
result, for it will be seen that in following the triangulation net 
westerly from the Richmond base the Dividing Range is crossed, 
and at all stations between Bindo and Buffalo the general in- 
fluence of the mountain mass is shown by an easterly deflection 
of the plumb line ; also in the southerly extension from the Lake 
George base a similar deflection of opposite sign is shown, re- 
sulting from the triangulation being located on the eastern slope 
. of Monaro with the high mountain mass culminating in Mount 
Kosciusko, the highest point in Australia, lying to the westward. 
Pp 
