44 Wadsworth — Simple and Accurate Cathetometer. 



In the new form of instrument the best position for the object 

 and the comparison scale is about 90° from the axis of the tele- 

 scope or in the direction, 0, fig. 2. If we suppose the, object 

 and scale 15° from one another, and symmetrically placed on 

 the two sides of the 90° position, we have for 6 and d 1 respec- 

 tively 90°±7i° or 97i° and 82J°. Hence, 



A = 0-26 r sin a^. m \ra (3) 



and A'= (4) 



The general equations (1) and (2) show that care in leveling is 

 only necessary in the vertical plane perpendicular to the line 

 of sight, i. e. in the new form the plane parallel to the axis of 

 the telescope ; in the usual form the plane at right angles to 

 that axis. Hence if the greatest accuracy is to be attained 

 with the ordinary cathetometer the usual telescope level should 

 be placed at right angles to its customary position (or perhaps 

 better still, a second level added in that position), so as to at 

 once call attention to any error of adjustment in that plane. 

 It is strange that this rather important fact should have been 

 overlooked in previous designs. 



The actual magni tuple of the error in measurement, due to 

 an error in leveling, is, however, always small, unless either the 

 distance of the object from the telescope is considerable, or 

 the difference between the angles 6 and d r is larger than 60°. 

 If a — 5" and d—d 1 = 15° as in (3), the error, J, for objects dis- 

 tant f M. from the axis of rotation, would be about *003 mm or 

 about the limit of accuracy of setting with the best cathetom- 

 eters under the best conditions. 



With a good level sensitive to 5" per division (the best cath- 

 etometer levels are from two to three times as sensitive as this), 

 there is no difficulty in setting by reversal to within less than 

 jr div. or V' ', reducing the error under the above conditions to 

 about Woo mm - 



This shows that we may very considerably increase the 

 angular difference d—d v without introducing any appreciable 

 error. For when this difference is 60° the value of A is only 

 twice the above values or about *O01 mm in the last case, a quan- 

 tity quite negligible in comparison with the errors of setting. 

 This indicates another method of using this new form of instru- 

 ment to good advantage, i. e., the method of superposition of 

 object and comparison scale. To accomplish this the mirror 

 A is half silvered and the scale is viewed directly through the 

 unsilvered half in the direction o' (see fig. 3) the object being 

 at the same time seen by reflection from the silvered half in 

 the direction o" * In this case we may make the measure- 



* The mirror should be half silvered horizontally, i. e. the line of separation of 

 the silvered from the unsilvered portion should be parallel to the axis of rotation, 

 both because the maximum resolution is required in a vertical direction and 

 because, as will be seen later, this management is the better adapted to the use of 

 certain forms of micrometer. 



