76 H. G. Fourcade.—On the Repetition of Angles. 
It is, nevertheless, very improbable that repetition will again 
supersede reiteration for work of the highest precision in which 
first-rate instruments are used. When £ the probable error of a 
reading becomes sma!l compared to a the error of a bisection, there 
is little difference in theoretical accuracy between the two 
methods, especially if account be taken of the rapid elimination, by 
the ordinary one, of the systematic errors of graduation. ‘There 
remain important advantages on the side of the present method: 
it gives the means of better eliminating, by multiplying the obser- 
vations on different days and under different conditions, the usually 
much more serious errors introduced by the atmosphere, and since 
with the perfection of modern instruments the terms involving 
@B become nearly as small or smaller than the terms involving a, it 
usefully allows of several bisections with an eye-piece micrometer 
for each reading of the circle. 
But with smaller triangles, when the accuracy aimed at is well 
below the limit practically imposed by the presence of the atmosphere, 
and a number of observations are still required with the instrument 
used, repetition often affords the quickest method of measurement 
of the horizontal angles. When the instrument is small and the 
error of a reading large in consequence, the advantage may become 
very great. With the theodolite for which gq = +080 and 
B= +604 the probable error of an observed angle was +0*49 
with ten repetitions and +028 with twenty, and these values were 
borne out by the smallness of the corrections required to satisfy 
the geometrical conditions of the triangulation. To have obtained” 
the same accuracy by reiteration 150 duplications in the first case, 
and 490 in the second, would have been needed. As accurate work 
can be done only during a few hours a day, it is seen that a precision 
which was attained in a single visit to each station would have been 
practically out of reach with reiteration. | 
The method of repetition may thus be often employed witl 
advantage in the secondary triangulation of a country. It ougkt, 
in particular, to become valuable to ordinary surveyors who are 
ealled upon to provide, by triangulation, a framework for extensive 
property or topographical surveys, because it allows of the use of 
small instruments for the most accurate work likely to be required 
in practice, two material axes without play, and strong springs to 
the tangent screws, being all that is essential to render a theodolite 
suitable for the application of the principle. 
