102 G. H. KNIBBS. THE THEORY OF THE 



Constant and periodic errors and their effects. 



The successive readings of the circle entered in column 2 of I., 

 being the means uf verniers 180° apart, * any error of eccentricity 

 as between the centre of the vernier-axis, and that of the graduated 

 circlt., is entirely eliminated. To rigorously determine the mag- 

 nitudes of the accidental errors of reading and pointing, however 

 it is further requisite that corrections be applied for any other 

 periodic errors entering into the results. Of these some are 

 peculiar to the repeating method and others though not so, require 

 to be noticed. A general investigation for the discovery of 

 periodic errors of graduation might with advantage, in some 

 instances, be made in the usual way,f but the graduation of 

 modern instruments is frequently so excellent that, especially when 

 resorting to repeating measures, this may be held superfluous. 



Of the strictly constant errors that demand notice, that of 

 variation of the line of collimation by changing of focus, and that 

 of collimation are perhaps among the most important. 



The former is generally very slight except for great differences 

 of focus. It may be measured by observing an angle with the 

 face of the instrument in reversed positions. Denoting it hy f, 

 the number of measures with the face in one position by m, those 

 in the other by m', and the total difference between the multiple 



measures by d, its value will be: — f= , (39). This 



assumes the measure to be made with the altitudes of the objects 

 zero, when f is to be regarded as its horizontal value. If the face 

 of the instrument be reversed for each successive measure in actual 

 practice, this error will be eliminated from the result, as also will 

 the collimation error, (c, treated of hereafter) provided 2m - 1 

 measures are taken and reduced as in II. % If the value of the 

 angle is obtained, however, by dividing the final reading by the 

 number of measures, that number should be even, to eliminate 

 both /and c. 



The error due to focussing is strictly, though it is not convenient 

 so to consider it, a collimation error, for the latter differs in this : 

 it has no effect upon angles when the altitudes of the objects 

 between which they are included are equal. When they are not, 

 putting m for the real error of collimation and c for the error it 

 introduces into the measure of an angle, c = m (sec (3 l - sec |3 2 )... 

 (40). § As c and f enter as combined constants into the 



* It may be observed, en passant, that the probable reading error 4'44' 

 is consequently that of the mean of the two verniers. 



f By the application of the theorems relating to periodic functions. 

 X Refer back to note respecting the 2m - 1 rule. 



§ The mode of measuring the various errors of instruments has been 

 treated in an article on the Rigorous Examination of Theodolites, by the 

 writer, published in "The Surveyor," the Journal of the New South 

 "Wales Association of Surveyors, Vol. i., No. 10. 



