REPETITION OF ANGULAR MEASURES WITH THEODOLITES. 89 



negative, as when their unequal illumination produces a phase or 

 constant error which tends toward the illuminated side, and 

 although these constant errors for each object, so, or otherwise 

 produced, may not be mutually eliminative, no inaccuracy will be 

 introduced by neglecting to recognise their existence when discuss- 

 ing accidental errors. For all constant errors must be treated 

 independently by appropriate investigations, determined, and 

 then eliminated by corrections ; and it is not proposed in this 

 article to examine any errors except such as are either peculiar to 

 the system under consideration, necessarily coexistent with them, 

 or such as must perforce be argued in this connexion. 



Probable error of Pointing. 



If errors of pointing be determined or expressed by their angular 

 value measured at right angles to the line of sight, which in fact 

 is the most legitimate way of estimating their magnitudes, that 

 part of the error which influences angles of azimuth is the hori- 

 zontal component only, or to express it otherwise, the distance of 

 the intersection of the wires in the reticule of the instrument from 

 the vertical line through the image of the object, measured at right 

 angles to that line. As the effect of this component varies with 

 the altitudes of the objects, (1) and (2) require modification to 

 make them generally true, being in view of the preceding qualifi- 

 cation restricted in their present form to the case where the 



altitudes are zero. In this we may substitute for (3), + — l. (4) 



for the errors p" and p' will, generally, each be subject to the 

 ordinary laws of frequency of accidental error, and have the sam e 

 measure of precision, or modulus. Being in their nature compen- 

 sating or mutually eliminative (equally likely to assume equal 

 positive and negative values) they may be discussed by the ordinary 

 theory of errors. Putting therefore, p o for the probable error of 

 a single pointing, the probable value of (4) by that theory is 

 ± Po^§ = ±Po v| or + 1>41 4 j£ (5), so that if ^ o can be 



evaluated, the probable error due to errors of pointing may be 

 ascertained for any number of measures. For example if p o be 1 ", 

 nine repetitions would be subject to a probable error of 0."47, viz. 



r"414 l"x V(2x 9) 



VO 9 



The effect of errors of pointing varying in their influence upon 

 azimuthal measures with the altitudes of the objects viewed, if the 

 latter be denoted by /3i and /3 2 , and p o signify as before the proba ble 

 horizontal error of pointing, its azimuthal values will bep o sec j3 x , 

 p o sec /3 2 (6); (4) cannot therefore be rigorously substi- 

 tuted for (3) in the general case, for the moduli of the errors are 



not identical, and thus (3) becomes + — E 2 -l El — L (7) 



n 



