REPETITION OF ANGULAR MEASURES WITH THEODOLITES. 103 



results, their sum may be denoted by k, the sign of which will be 

 reversed with a reversal of the face of the instrument. In I. 

 therefore, since the readings marked 1, 3, 5 etc. contain k, and 

 those marked 0, 2, 4 do not, the addition of the seconds in the 

 third column of the respective readings shewing seconds only 

 indicates that a -k — 466" -f- 9 and a + k — 559" -^ 10, k therefore 

 being + 1'05". The result is simply illustrative of a method 

 legitimate only with a very large number of such measures and 

 in this example has no intrinsic value. It may more accurately 

 be found, by the process to which (39) applies, substituting k for 



f, thus similarly k = , (41) the general formula, (39) being 



merely a particular instance where fi is 0°. c and f may be 

 differentiated out of this result by means of (40) when m is known.* 



If the pivot axis of the telescope be not perfectly adjusted so as 

 to make the plane of revolution of the latter vertical, a constant 

 error, also changing its sign with change in the face of the instru- 

 ment, will enter into the results. This too is included in the 

 general formula (41). 



Turning now to periodic errors, one of these, essentially peculiar 

 to the system of theodolite repeating-measures, is that due to want 

 of parallelism between the azimuthal axes, the outer and inner. 

 This error operates in the following way : — as the relative positions 

 of the two axes change, the plane of revolution of the telescope 

 departs from the vertical, an amount determined by the inclina- 

 tion of the axes, and the sine of the angle through which the axis 

 is turned, f If therefore j3 x and /3 3 be not zero, each reading will 

 be subject to a correction, without which the result will be more or 

 less in error (except in a particular instance hereinafter mentioned) 

 and the investigation of the probable value of the reading and 

 pointing errors, and of the absolute amount of the focussing and 

 collimation errors, is impossible. If I denote the elevation of the 

 right hand side of the pivot axis of the telescope (looking along 

 the telescope), and z the error of azimuth caused by this defect, 



z = I (tan #! - tan (3 2 ) (42) or if this inclination vary as it does 



in repeating, z = l x tan/3 x - l % tan /3 3 (43). The angles I 



may be read for variations caused by the differential revolution 

 of the axes, by means of the level upon the vernier plate, or 

 with the striding level ; or may be computed when the inclination 

 i of the axes, and the direction of the plane passing through the 

 the two is known (y). If then in repetitional measures the outer 



* It is convenient to regard the error of collimation at the principal 

 focus (or when the instrument is focussed for infinite distances), as the 

 true collimation error ; and to denote the variations from this value at 

 different focal distances, errors due to focussing. 



t The method of measuring this error is described in an article pre- 

 viously quoted. 



