104 G. H. KNIBBS. THE THEORY OF THE 



axis, as ib is desirable it should, be made vertical, the series of 

 errors in the measures of the angles will be as follows : — l x tan /3 X 



- l 2 tan /3 2 ; 1 2 tan /3 X -l s tan£ 2 ; l n fi x - / r . + i/3 2 (44). 



But l x etc., are functions of i, y and a, thus ; — l x =i sin y, L 2 =i 



sin (y+a), Z 3 = t sin (y+2a) etc (45).* If a happen to be a 



submultiple of 360° ; say - -, the last terms in (45) become i sin 



{'y+(w — l)a} and i sin y, (46): and since (44) may be written 



in this case, — (l x -\-l 2 -\-l 3 -±-...-\rl n ) (tan/3 l - tan/3 2 ) (47), 



inasmuch as l n +i = li, and as the sum of the series i sin y, 



i sin fy -f (n — 1) al =0 whena = -— , it, (44) becomes zero: in 



other words the errors are eliminated. If the telescope be reversed 

 after each measure, the desirability of which has been previously 

 indicated, (47) will take the form : — (l x - l 2 + l s - ... +l n ) (tan /3 X 



+ tan £2 ) (48), so that if n be an even number the last 



term will be - I, and the series may be written : — i sin y + 



i sin |y+(a + 7r)) + + isin\y+(n - 1) (a+w) ] (49), the sum 



of which (to n terms, a being — ) is zero, a result obvious also from 

 the fact that the + terms in the first factor, and also the - terms, 

 differ when expressed as in (45) by -„ — , each taken separately 

 being therefore zero. 



The result of this investigation may be thus expressed : — If the 

 pivot axis of the telescope be adjusted so as to be at right angles 

 to the inner azimuthal axis, and in taking a series of repeating 

 measures, the outer azimuthal axis be made vertical, the effect of 

 inclination between the latter is eliminated in the final reading, 

 if the angle measured be a submultiple of 360°, whether the 

 number of the series be even or odd, provided the face of the 

 instrument be not reversed, and if the reversions be successive 

 without this qualification should the number be even.f The 

 maximum error will occur when na is 180°. It is evident that if 

 na approximate to 360° the error will be correspondingly small. 



This result is very useful in observations for determining the 

 true meridian by azimuths of the pole star, or of circumpolar 

 stars, owing to whose altitudes the errors last considered enter 

 into the measures with large factors. Fortunately for the general 

 precision of geodetic observations, this remark does not hold good 

 in reference to the measurement of the angles of geodetic triangles. 



* y denotes the angle measured from the plane passing through both 

 axes to the initial reading. 



f The (2m- 1) rule is open to objection in view of this result, as, if 

 reversion of face take place at each successive reading- to eliminate 

 collimation and focussing errors, a small error will be introduced through 

 imperfect elimination of axial error. It is easy to decide in specific cases 

 which course to adopt. 



