REPETITION OF ANGULAR MEASURES WITH THEODOLITES. 99 



III., which also serves for readily finding the probable error of 

 any result by the method of I., when p and g are known, and for 

 deciding for any given variations in the values for these last 

 quantities, what number of measures will produce an equivalence 

 of probable error, or afford results of equal reliability. 



In regard to the practical application of the Table, if the prob- 

 able errors of pointing and reading are ascertained (for each 

 particular instrument used) by a large number of observations 

 made under the atmospheric and other conditions generally 

 obtaining, the probable error of the result of any number of 

 repetitions, in which only the initial and final readings are noted 

 {or employed), may be very fairly defined on the basis of these 

 ascertained errors. Thus the mean of nine measures with a prob- 

 able pointing error of 04", and graduation of 2*0" (or five times 

 the former) would be 037", viz. 04 x 0'92, this last quantity 

 being taken out in table under 9 and opposite 500. 



As another example of its use, suppose that (being necessary to 

 omploy in the measures of the angles of a geodetic survey two 

 theodolites, in one of which a number of investigations indicated 

 that p and g were respectively 0'2" and 1*6", and in the other 0*3" 

 and 3"0") it was desired for the purposes of ready reduction, to 

 " repeat," when observing, that number of times which would 

 make the work of each instrument equally reliable. To elicit the 

 required numbers from the table, it will be necessary to select in 

 line 800, (100 X 1*6 -f- 0*2) one or more values which when multi- 

 plied by 0-2 will equal values in line 1000, (100 x 3-0 v 0"3) 

 multiplied by 0*3. Multiplying out the lines mentioned (by 0'2 

 and 0*3) and comparing, it w T ill be seen that the result given by 

 eight repetition measures with the former instrument has the 

 same probable error as that of fifteen with the latter, viz. 0*30". 

 If this were considered an excessive number, four and eight 

 (0-58" and 0-55"), or five and nine (047" and 0-50") might be 

 employed, or vice versa fifteen and thirty (047" and 046"). 



If the original evaluation of p and g (as of course it should) be 

 founded upon observations reduced as in I. and II., the determin- 

 ation of the relation of the measures with the two theodolites will, 

 then be a legitimate empiricism, and a reasonable estimate of the 

 probable error of the measures can, at least approximately, be 

 made. 



The Table exhibits the magnitudes of the probable errors for 

 different numbers of repetitions, the probable error of pointing jt? 

 being regarded as 100 and constant, for errors of reading and 

 graduation g, from 200 to 2000. Thus if p represent 1" the 

 quantities will represent hundredth seconds. To find the probable 

 error of a result of n measures, multiply the quantity under n and 



