REPETITION OF ANGULAR MEASURES WITH THEODOLITES. 91 



microscopes, or in the dividing of the graduated circle, or arising 

 from mechanical imperfections in the construction of the instru- 

 ment such as want of parallelism between the axis of the graduated 

 circle and that carrying the verniers, or constant errors, such for 

 example as that of collimation, must be examined, measured, and 

 eliminated by corrections, and for the purpose of the following 

 investigation these corrections are supposed to have been applied 

 and the several results to be, therefore, free from such errors. 



Denoting the errors under consideration by the letter g and 

 remembering that not only was the first setting at zero subject to 

 an error of this character, but also every subsequent reading, the 

 terms in (2) properly varied to represent the actual instead of the 

 true readings, will be as follows : — 0°, a— Pi +p 2 ~ 9i +9zi %a — 



P1+P2 -Ps+Pi-91+93, ,7ta-p v -\-p 2 - +p» n - 



0x+gn+l (13). 



With regard to the final quantity in (13), the terms following 

 na constitute the total actual error, the first part of which has 

 already been symbolically ascertained. When the value of a is 

 found by dividing this final reading by the number of repeats, the 

 only errors of reading that enter into the result are the primal 

 and final ; the probable error due to these is therefore that of 



— ^ — yn+i . wn j cn< putting g for the probable value of g x or 

 n 



p m+h is ± g -±f or ± 1-414 & (14). 



Combining (12) and (14), the probable error of the mean of n 

 repeats is : — 



±v{|r+^-(sec 2 /3 x +sec 5 ?0 a )} = ±J v /{2g 2 +m ^( S ec 2 / 3 1 +sec 2 /3 2 )} 



(15) and this, when /? t and /3 a are each zero, becomes : — 



t™ <&.' +?V?) ;••• (16)* 



From (15) and (16) it is evident that to find the probable 

 error of the result of repeating measures, the probable errors of 

 pointing and of reading must each be ascertained ; but if the 

 number be very large the result may, in some instances without 

 serious error, be assumed to be affected by errors of pointing only. 

 In determining the number requisite to warrant such an assump- 

 tion the relative magnitudes of g and p furnish the necessary 

 criterion. Thus for example if nji 2 =5(/ 2 , the probable error will 

 be varied about one tenth, if 10#-, one twentieth, from the correct 

 value. 



A better result will, however, always be obtained by (even 

 roughly) estimating the value of g and including it. For example 



the error due to imperfect construction of the microscope, and that of 

 the observer's reading is doubtless the least. 



* See Table IV. 



