Simple and Accurate Cathetometer. 127 



e' =• r sin at! sin 6 ; 



and for the corresponding errors in comparison of object and 

 scale : 



A =e — 6 X = r sin « (cos 6 — cos Q x ), . . . (1) 



A' = e'-e 1 Wsina / (sin0-sin<9 1 ). ... (2) 



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

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

 telescope, or in the direction o, 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 

 0i respectively 90° ±7£° or 97^° and 82^°. Hence 



A = 0-26 r sin aS'im . .... (3) 



and A'=0 (4) 



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

 levelling 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 tele- 

 scope 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 magnitude of the error in measurement, due to 

 an error in levelling, is, however, always small, unless either 

 the distance of the object from the telescope is considerable, 

 or the difference between the angles 6 and 6 1 is larger than 

 60°. If a=5" and (9-^ = 15° as in (3), the error, A, for 

 objects distant JM. from the axis of rotation, would be about 

 •003 millim., or about the limit of accuracy of setting with 

 the best cathetometers under the best conditions. 



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

 cathetometer levels are from two to three times as sensitive 

 as this), there is no difficulty in setting by reversal to within 

 less than £ div. or 1", reducing the error under the above 

 conditions to about -juoo millim. 



This shows that we may very considerably increase the 

 angular difference — 6^ without introducing any appreciable 

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

 twice the above values or about *001 millim. in the last case, 



