KEPETITION OF ANGULAR MEASURES WITH THEODOLITES. 97 



errors of pointing and two of reading, but the errors of reading 

 are different in each instance, and two errors of pointing are in- 

 cluded, not included in that preceding; two likewise being excluded. 

 Thus the measures are perfectly symmetrical as regards their 

 errors, and are varied so as to secure whatever advantage the 

 utilization of every reading taken affords ; an advantage counter- 

 balanced only by the repeated inclusion of some only of the errors 

 of pointing. But when those errors, as compared with the former, 

 are small, it is a priori evident that the advantage is considerable. 

 The limits of advantage will form the subject of inquiry later on. 

 In this method, a rule for which is now given, each reading has, 

 as evidently it should, precisely the same influence on the result. 

 Taking an odd number (2m - 1) of measures of the angle the value 

 of which is required, subtract the first or initial reading from the 

 (m + 1)^, the second from the (rn + 2)th, and similarly to the rath 

 from the 2mth, or final reading, adding to each difference the proper 

 multiple of 360° : the sum of the differences divided by m 2 will be 

 the mean value sought* 



If the number of measures be even, 2m, subtract the first read- 

 ing from the (m+l)th, and so on, adding as before the proper 

 multiple of 360°, the final subtraction being the (m + l)th from the 

 (2m + l)th, and divide the sum of the differences by m(m + l). In 

 the latter case the (m + l)th reading has no influence whatever 

 upon the result, upon which fact the dictum, that an odd number 

 of measures should be taken, is based. 



Evaluation of probable errors of residts. 



The probable values of the errors of pointing and reading having 

 been disclosed by the preceding investigation, the application of 

 formula (16) immediately gives the probable error of the result 

 presented by the method of I. Thus, (1-4144-19) x v{ 19-73" + 



(19 x 0'55 2 )} = + 0-38", (the first approximate values for g o and^? o 

 give + 0-36"); and the result may therefore be expressed, 82° 59' 

 52-9" + - 4", a result evidently agreeable to the indications given 

 in column 6.f 



In ascertaining the probable error of the value found in II., it 

 is to be remarked that it is influenced by thirty-eight (2x19, or 2n) 

 errors of pointing and twenty (19 + 1, or n+1) of reading. It is 

 necessary to have regard however, not only to the number of errors 

 but also to the way in which they enter into the result. The 

 following series will represent the actual quantities (the multiple 



* The elimination of collimation and of focussing error by this rule is 

 adverted to later on. 



f A Table (III.) is given hereafter by means of which the probable 

 error may be readily formed by inspection. 



G— August 6, 1890. 



