30 



indeed a real objection against this mode of placing the division, 

 and almost overbalances the great stability, which is attained by 

 fixing the microscopes to the pier. The expansion of the radii or 

 arms of the circle brings the limb nearer to the object glass of the 

 microscope, and that of the limb increases the magnitude of the 

 interval betvi^een two divisions. The distance of the object-glass 

 from the micrometer also increases, but this is nearly counteracted 

 by the expansion of the micrometer screw. 



Referring to the formulae of the last section we have, neglecting 

 the higher powers of h, and remarking that I becomes, by the ex- 

 pansion of the limb 1 [l+«r] where « is the expansibility of brass 

 for one degree, and r the excess of temperature above that at which 

 the microscopes were adjusted 



but h = — r s r, therefore 



/ — 1" error in seconds f >* . ''F* 



r r rF* 1 



I 300 



a quantity much greater than in the former case, for in a five feet 

 circle it is = 0"47xr; it does not receive the least compensation 

 by increasing the number of readings, and there appears no remedy 

 but that of registering the run of the microscopes at each observation, 

 or until a table of its amount at different temperatures can be 

 formed. If indeed they were attached to a circular frame similar and 

 concentric to the circle itself, this error would be corrected, and with 

 it probably those which are noticed in the following sections : the 

 microscopes would be less stable, but it would be easy to apply a 

 level or telescope to keep watch on them. 



4. l^e unequal expansion of the arms of the circle deforms its 

 figure, and that of the limb disturbs the equality of its divisions; 



