OF THE UNITED STATES. ^yO 



whence, 



S= 





and since by the supposition, ('2n+i)B=Zr, (2n+^)s=2T — \a 

 - — \^. the two sines in the parentheses become identical, 

 and consequently S=0. 



It follows tlieiefore, that wliatever be tlie number of equi- 

 distant readings into which the circumference is divided, the 

 indiscriminate mean of all the readings will give the true an- 

 gle at tlie centre of the division. It is evident that for three 

 microscopes, ?;=i, so that (2«+l)/3=360°=3^. 



The same circumstance which occasions the eccentricity of 

 an instrument, may also cause the axis of motion not to be 

 perpendicular to the divided plane of tlie circle. The a.xis be- 

 ing placed vertical, by the adjustment of the instrument, the 

 plane of motion, thus horizontal, will not coincide with that 

 of the divided circle upon which the readings are made, and 

 will require a reduction to the imaginary horizontal plane, 

 ^vIlich will l)e exactly analogous to the reduction of the ec- 

 liptic to the equator, and may be determined by the formula 

 given for that purpose. 



It is evident that in changing the position of the instrument, 

 so as to make the legs successively change their places, the 

 plane of the circle will be placed in the same symmetrical 

 j)ositions with respect to any angle measured upon it, as was 

 the case in the readings of the angle ; and the angles will 

 lequire successive reductions corresponding to the same 

 numlier of symmetrical arcs. 



Tlie indiscriminate mean of these an^jles, observed in all 

 these positions, will again l)e the true horizontal angle cor- 

 rected for the want of perpendicularity of the axis upon the 

 divided liml). 



Plate VI. tig. .3. — Let ad l)e the inclined limb of the di- 

 vided circle, ac the horizontal i)lane, a the point of intcrscc- 



