158 REPORT OF THE CALIFORNIA EARTHQUAKE COMMISSION. 
At Crystal Springs, where the lines are much longer, the closing of the triangles was very 
good. The correction for each angle, in order to make the sum of the angles of each tri- 
angle equal to 180°, was on the average only about one second of arc. This goes to indicate 
that this instrument, when properly used, is capable of excellent results. At Crystal Springs 
a least square reduction of the observations has been made, but the angles and distances 
thus computed are almost identical with those of the original computation. 
At Olema, where the three lines having the least differences of level were measured, the 
diagonal between the northwest monument and the southeast monument (being best suited 
for computation) was taken as a base for computing the other two measured sides. The 
means of the computed and measured distances of these two sides, together with the direct 
measure of the above-mentioned diagonal, were taken as the best measures for computing 
the unmeasured sides. The lengths of the three unmeasured sides, therefore, depend not 
only upon the observed angles, but upon the lengths of the three bases, as indicated above. 
This method gives the measured distances and observed angles about equal weight, the 
angles being corrected for each triangle according to what is known as the “field adjust- 
ment.’’ As above noted, however, it is very doubtful if the angles are entitled to as much 
weight as the measured distances, and hence, it was decided to retain the exact values of 
the three measured distances, and make a “least square” adjustment of the angles of the 
quadrilateral to correspond. 
The three measured sides being assumed as fixed, the three angles of the triangle N.W. 
Mon., 8.E. Mon., S.W. Mon., can each have but one value, and these values have been com- 
puted from the three sides. These three sides and the corresponding angles remaining fixed, 
an adjustment is made between the remaining angles and the three unmeasured sides, so 
as to fulfill all the geometrical conditions, giving at the same time the most probable values, 
according to the theory of “least squares.” 
Abstracts of horizontal angles. — In the abstracts of horizontal angles tabulated below, 
the first set of angles given under the heading “ Observed”’ are the means of angles taken 
directly from the original records. The column headed “Field Adjustment” shows the 
angles as they appear in the field computations after the angles of each triangle have been 
corrected to sum up 180°, giving the same correction to each angle in a particular triangle. 
This adjustment, which is the usual one made in the original computations, does not account 
for the other geometrical conditions required for the rigid solution of a quadrilateral, but 
when the errors in the angles are small, the resultant distances, especially if short, will be 
very near the truth. The column headed “Least Square Adjustment” shows the angles 
computed so as to fulfill all the geometrical conditions, giving their most probable values 
according to the theory of “least squares.” 
In future measurements, it will not be necessary to repeat the base-measurements unless 
the angles show some change, for by occupying all the stations, any change that could be 
detected by tape measurement will at once show in the angles. When, however, the angles 
indicate any change, then a remeasurement of at least one line will be necessary. 
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