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Methods of Astronomical Observation, 185 



comprehending those usually employed in recording obser- 

 vations, while at the same time the sliding vane must be 

 moved till its mark again coincides with the intersection of 

 the cross wires in the telescope, still clamped to the circle, 

 and the number of divisions on the rod which it has passed 

 over to thousandths, or, at least hundredths of a foot, by 

 this motion must then be recorded. 



4. Measure the horizontal distance with great care 

 between the centre of the circle and the levelling rod. 

 These afford data for computing trigonometrically the value 

 of the divisions of the scale of the level. 



5. To investigate formulae for this purpose, let R" be the 

 length of an arc equal to the radius in seconds, D the 

 horizontal distance, d the distance passed up or down by 

 the vane, A" the arc in seconds subtended by d, at the 

 distance D then by the principles of trigonometry, 



A' = 5:^ (1) 



If L be the length of a given number of seconds, a" on 

 the scale of the level, and r the length of the whole run in 

 the same measure as D and d, 



^ = R" X d ^^^ 



Indeed, if any four of the fiye quantities, D, d, r, a", and 

 L be known, the value of the fifth may be found by trans- 

 forming the preceding equation, thus : 



If n be the number of divisions in the run of the level, 



« = B-^n • • (4) 



If P be the radius of curvature of the level, 



R" X L R " X r ,_. 



p = ^^^ = -^^ (^) 



Examples for the use of these formulae. 



1 . The cross wires of the telescope of an astronomical 

 instrument, at the distance of 250 feet from a levelling rod, 

 moved over two inches in a run of the bubble through an 

 inch and a half, by turning the feet screws in the direction 



