REPETITION OF ANGULAR MEASURES WITH THEODOLITES. 



101 



The probable error in (26) due to errors of reading being + c/ o 

 V(2m), which divided by m 2 expresses that of the mean result by 



II., viz: — +g o J— § (32), the combination of this with (30) gives, 



V q * {2m2 ^l +6g ° 2 (33)* that is the probable error of that result. 



That of the method of I. is, remembering n = 2 m - 1, */ q ( ™ X* g ° 



' © > (2m— 1)- 



(34). To find the limit, (33) and (34) must be taken as 



equivalents. Squaring and removing the fractional form gives 

 the equation :— (2m- l) 2 {^ 2 (2m 2 +l)+6^ o 2 } = 3m 3 {# 2 (2m - 1) + 

 2g »■> (35) and this reduced is, 3»? = ^-5m^ + ^-\m+l ,3^ 



If any numerical or rational values be assigned to # o and #,(36) 

 may be solved for m, or for 2m - 1, that is for the number of 

 observations producing equality of probable error, and beyond 

 which the system II. is inapplicable or disadvantageous. 



In the following table, IV., the argument is -, the probable error 



of reading divided by that of pointing, p being \. Solving 



algebraically, one of the roots, it will be observed, is 1 ; a result 

 true for any value of the left hand member of (36), and the 

 physical interpretation of which is obvious. As only integral 



values are required for 2m - 1 , the solution is simple after - = 3, 

 when ~ will be 27, and the right hand member of (36) may be 

 written 2m + 3 + - 1 , the last term serving for a second approxima- 

 tion : and after - = 4, (36) may be still further simplified • for 

 putting c for that quantity, and remembering that -—■ = 3c 2 , and 



..(37) 



3c 2 -3 



that results are required but to the nearest unit, in — 



orn = 2m -1 = 3c 2 -4 (38). 



If then n be greater than 3c 2 - 4 the process exhibited in II. 

 should not be followed, and this is the required criterion, tabulated 

 in IV. 



IV. 



Limits of 2m - 1 Rule. 



J7 



P 



n 



P 



n 



1 

 P 



n 



9_ 



P 



1 

 n 



9 



P 



n 



2-2 



5 



31 



24 



3-6 



35 



4-5 



57 



7 



143 



2-4 



9 



32 



26 



3-7 



37 



5-0 



71 



8 



188 



2-5 



.13 



33 



29 



3-8 



39 



55 



87 



9 



239 



2-8 



18 



34 



31 



3-9 



41 



6-0 



104 



10 



296 



3-0 



22 



35 



33 



4-0 



44 



6-5 



123 



11 



359 



* A table similar to III., might be constructed with the same arguments 

 using formula (33) instead of (16) for use with this method. 



