88 G. H. KNIBBS. THE THEORY OF THE 



the movement of the azimuth circle, (the revolution of the outer 

 axis) until the cross wires appear to be exactly coincident with the 

 object viewed. Releasing then the inner axis, to which the 

 verniers and the telescope are invariably fixed, and revolving it ; 

 the latter is now similarly directed and adjusted on the second 

 object, clamped in position, and the arc, registered by this move- 

 ment upon the graduated circle, read off. The outer axis is again 

 released and adjusted as before on the first object, the verniers 

 remaining at the same readings, as when on the second. After 

 clamping in position the release of the inner axis and re-direction 

 of the telescope upon this last, scores the angle between the objects 

 a second time on the circle. This operation is repeated as often 

 as deemed advisable, and consequently if the directions and read- 

 ings were perfect, and if the mechanical construction of the instru- 

 ment were absolutely faultless, the readings would be 0, a, 2a, 

 3a, na, a denoting the angle between the objects.* 



Errors of Pointing. 



The adjustment of the cross wires in the reticule of the instru- 

 ment on the image of the object is always, it is hardly necessary 

 to say, but an approximation, and if the error of the attempt, 

 technically called an error of pointing , be denoted by p, instead 

 of the telescope having been turned through the arc a each time, 

 it is really turned through the arcs : — 



a-p 2 +p 2 , a-ps+pt, a-Pzn-x+Pzn'- (!) 



in which expression p 1 denotes the error made in pointing to the 

 first object, /?._, to the second, p s to the first object the second 

 time, p± to the second object the second time, and so on; p being 

 either positive or negative. The true readings on the arc are 

 therefore: — 0; a-p 1 +p 2 ; 2a — {p l +p :i ) + (pz+p*)', na- 



(Pl +P.3 + ••• +P2Xl-i) + {P2 +P+ + •■■• +P2n) : - ( 2 ) SO that 



if there were no graduation errors, and if the graduation could be 

 read perfectly, the error of the measure of a as obtained by the 

 usual method of dividing the final reading as above, by the number, 



n, of repetitions, would be +— - — — — (3), in which the 



n 



first term in the numerator denotes the final term (in brackets) in 



(2); and the second, the one preceding that. That is to say, the 



numerator in (3) represents the algebraical sum of the whole of the 



errors of pointing. 



Errors of Phase Irrelevant. 



Now, although the probability of the actual errors of pointing 

 at either object may not in certain instances be that of their being 



* By making therefore n large enough the effect of any error in the 

 graduation of the circle may be diminished to within any desired limit. 



