OP THE UNITED STATES. 6U t 



It is evident that (he series in this sum are similar to that be- 

 fore considered, being the sums of sines of arcs in arithme- 

 tical progression, limited by the sum of the &'s being equal 

 to the circumference of the circle, which makes their sum 

 =0, and proves the indiscriminate mean of the angles ob- 

 served in the symmetric positions of the instrument to be 

 the accurate iiorizontal angle. 



In an eccentric instrument, it is of course impossible to 

 make the microscopes measure exactly in all parts of the 

 division, but the above shows that if they arc adjusted in 

 any one position, tlieir measure will be corrected Uy the 

 changes of position of the instrument, without having re- 

 course to any other means. 



EiTors may also arise from a want of horizontality in the 

 axis of the instrument. It is proper therefore to adapt the 

 method of observing so as to correct these errors. But such 

 errors are easily corrected in this instrument, by observing 

 with the telescope in two positions diametrically opposite to 

 eacli other. 



In Plate VI. fig. 2, let ab be the horizontal line in which 

 the axis should be, and tp the section of the (rue vertical 

 plane wliich (lie telescope should describe. Instead of this 

 let the axis be inclined in one position of the instrument, so 

 that the telescope moves round the line a'b', and describes 

 a circle making with the vertical an angle /rf',= loi'^bcb'. All 

 the results of observations on objects taken in this plane will 

 require a reduction corresponding to this angle. Turning the 

 telescope so as to revolve through a semicircumference ho- 

 rizontally and vertically, and observing thci same objects again 

 without any change of the adjustments of the transit, the axis 

 will come in the direction a''b'\ and the plane of revolution 

 of the telescope will make with the vertical the angle t''ct= 

 t'd ; but on the side exactly opposite. All results of ob- 

 servations will require exactly the same correction as before 

 in respect to quantity, but they will be negative in respect to 

 the former, and the indiscriminate mean of the two will be 



VOL. n — n 2, 



