AN INVESTIGATION INTO THE EFFECTS OF ERRORS IN SURVEYING. 853 



Hence, when x is less than x , or greater than (180° + a; ), the error (S- T) will have 

 a positive sign, while for all values between x and (180° + « ) it will have a negative 

 one. 



Dealing firstly with the semicircle ENF, consider the point N as successively 

 occupying a large number of equidistant positions on the circumference of the semicircle, 



7T 



the angular interval between each being Sx. There will be j- of these positions, and, 



by means of (3), the angular error can be ascertained for each of them. 

 The average angular error over the semicircle will therefore be : 



2 i k 8in (2 +x ) + k sin (z "•'')} 



1 r+tan" 1 [( £ ^)tan ¥ ] 



7T 



Sx 



or, proceeding to the limit : 



■*-"[(g±g)*-a 



Average angular error over I _ r 

 the semicircle ENF ) ~ "^ 



\ L x • L 2 j 



= ifc\/( L ' + L - 2L ^ COsT ) 



(5) 



The same integration-process determines the error over the semicircle EZF, the 

 limits being, however, reversed ; therefore the result is the same as (5), but with a 

 negative sign. Hence the average error over the whole circle may be written — 



The average angular error due to \ 2r I / 1 1 2 cos T \ ,g. 



a displacement, r, in centre j ~ - V\/ \Lf Ljj LjL., / 



Thus, for a given value of r, the average angular error is a maximum when the 

 traverse angle is 180°, and a minimum when the traverse anale is zero. 



Section III. The Relative Effects of Errors in Centring and those of 



Sighting, Reading, etc. 



Assuming that a theodolite in good adjustment is employed, and that a method of 

 measuring angles is used by which the effects of the main instrumental errors are 

 reduced to negligible proportions, an angular measurement will be affected with errors 

 of sighting and reading the instrument, and also with other minor, and more or less 

 obscure errors — quite apart from that due to centring discussed in Section II. With 

 the exception of that of sighting, these errors are independent of the lengths of the lines, 

 and can therefore be taken as having a constant average value. 



The error of sighting is more difficult of treatment. It is very certain that the 



