1892. | Repetition of Angles. 69 
Summing these series, it is found that 
sin (a — np) sin (nx — 1) B 
2S. = (2 — 1) sin a — wane 
: sin (A — np) sin (xn — 1) 6 
28, == (n —1) sin A — eam ais ed cae 
B 
in which 6 —@> aud o = 4 = Bs 
and the correction # to the angle obtained by repetition is 
Uy + %+ s.. + x, 
7 : 
x 
°/f 
= — (2.S_ cot z — 2S) cot 2). 
It is seen from this formula that the correction will vary is 
magnitude and change in sign with different values of a, B, 7, and 
Z1, Z2 + ‘By reversing the direction of rotation, it is only in particular 
cases that the mean will be freed of error. The method due to 
Struve thus fails in ordinary cases, although it sometimes may 
reduce the error very considerably (for instance when A is nearly 
equal to S or 180° + - with little difference between z, and 22). 
But by making A into 180° + A, the sign of the correction is changed 
and the mean freed of the error due to the inclination between the 
axes without preventing the simultaneous elimination of the remaining 
instrumental errors by the method of reversal. 
In order to compare calculated corrections with actual discre- 
pancies, an angle was observed on the 20th Nov. 1890, at the station 
“ Roode” between ‘“ Spijoenkop” (— O° 51’ 20” app. alt.) and 
‘“‘ Hoogeberg ” (+ 5° 59’ 30” app. alt.) and two series, with the 
origins O° and 180°, obtained. This angle was selected because it 
gave the largest difference in zenith distance in the survey, and 
would, on that account, make the character of the errors most 
apparent. ‘The observations were : 
