OCTOBEE 23, 1896.] 



SCIENCE. 



619 



the Nadir will be ± 0^^.15 for one position, 

 =h 0^^.10 for both positions, 

 quantities which will not be diminished by an 

 increase in the number of observations. 



In the use of a movable circle, since new 

 divisions are brought under the microscopes for 

 the Nadir reading, each measure of declination 

 will be affected by the probable error g oi ± 



The mean of four observations would have 

 the probable error g of ± 0^^.10, whether made 

 in one position of the instrument or in both, 

 By increasing the number of observations the 

 movable circle would gain in accuracy over 

 the fixed circle, in so far as this source of error 

 is concerned. Thus, for eight observations, four 

 made in each position of the instrument, for the 

 fixed circle g would be as before ± 0^^.10, while 

 for the movable circle g would be d= 0^^.08. 



For differential determinations of declina- 

 tion the result, in both cases, will be affected 

 by the probable error of graduation for the 

 mean of the stars used as standard. If eight 

 fundamental stars are used, the probable error 

 g of the mean would be ±: 0^^.05, and the pro- 

 bable error g in the resulting declination would 

 be ± 0^^.16, from one observation. 



With the fixed circle in both positions of the 

 instrument the declination would have g equal 

 to ±0^M1. 



The movable circle would give g, 

 for two observations ±0^^.11, 

 for four observations ± 0''''.08, 

 and for eight observations ±0^^.06. 



The above treats of a movable circle shifted 

 for each night of observation. The custom of 

 changing the position at stated intervals will 

 have an advantage, only, in bettering the de- 

 termination of stars observed as standard over 

 long periods. 



In order to obtain a more complete compari- 

 son, it will be necessary to combine the effect 

 of graduation error with the accidental error 

 from other sources. The purely accidental 

 error may be assumed to be zfc 0^^.25 for one ob- 

 servation, including that of the determination 

 of the reference point, by either standard stars 

 or nadir readings. While it may be smaller in 

 some cases, this value can be used as a basis of 

 comparison. 



The probable errors of declinations, deter- 

 mined with fixed circle and with movable 

 circle, by both methods, would be predicted as 

 follows : 



Dif- 



Nadir. ferential. 



Fixed Circle : // // 



One position, 2 observations ±0.23 ±0.24 



4 " ±0.19 ±0.20 



8 " ±0.17 ±0.18 



Fixed Circle : " " 



1 observation in each position ... . ±0.20 ±0.21 



2 " " " " .... ±0.16 ±0.17 

 4 " " " " .... ±0.13 ±0.14 



Movable Circle : 2 observations... 

 4 



±0.23 ±0.21 

 ±0.16 ±0.15 

 8 " .... ±0.12 ±0.11 



With an increase in the number of observa- 

 tions the movable circle would gain slightly. 

 But, on the whole, it is doubtful whether the 

 gain is not more than compensated by the ad- 

 vantages of having a permanent relation be- 

 tween circle and telescope. Some of these 

 advantages become apparent when old observa- 

 tions are to be discussed for light upon some of 

 the refinements of investigation. This per- 

 manency and consistency give a definite basis 

 to work upon, when treating methods or results. 



As an individual illustration, since the usual 

 method of determining telescope flexure in- 

 cludes the sine term of the circle flexure, this 

 last does not need to be independently measured; 

 the effect of both may be determined, and their 

 laws studied together. With a movable circle 

 the flexure would need separate treatment for 

 each position. 



This comparison has been suggested, in part, 

 by a note in this Journal, in which the prefer- 

 ence for the movable circle is indicated ; and 

 the utility of the measurement of Division error 

 is questioned. (H. J., Science, Jan. 31.) 



The actual measurement of errors of gradua- 

 tion would, presumably, replace those errors 

 by the probable uncertainty of the measure- 

 ment. With the method employed for this in- 

 strument, the probable error of the measure- 

 ment of the 1° arcs is less than ±0.^^04. 



Since each division may be measured, as has 

 been done thus far, from two others, the probable 

 error can be kept within small limits by making 

 a sufficiently large number of measures. Practi- 



