29 



i — i' _i { 1 JP I 



( 1 F' -i ' 



^ ^"*^- X D^Tf + f.{D—Fr I 



The quantity h is the effect of excentricity in the direction of the 

 radius, or at the nth microscope. 



substituting which in the above expression, and integrating, we 

 should obtain the mean error of the n readings, if the 2 of such 

 a function of n could be assigned in finite terms. Developing in 

 powers of the cosine the value of I — F-i-l, it takes the form 



c COS. («a + J) — c*. cos.'^+&c. 



The odd powers disappear in n readings, but the even do not, for 

 each of them contains a constant, and as in this case the error 

 affects only one extremity of the arcs measured, they are not re- 

 moved by subtraction. There remains in the mean 



where the general term is (^ * (^~- • • ^~ ' • • ■^•) 



With microscopes similar to those of my equatorial, which have 

 F=2 and D = 3 inchess, the first of those terms is 408 T'x^^ at its 

 maxinmm. With a single microscope the total error is about 

 ld65h", or a fourth of what the same value of h would produce as 

 excentricity in a five foot circle. 



In circles divided on the plane, variable exentricity produces the 

 same error, though in a less degree, as also any deviation of the 

 limb from its plane. 



3. But a much more serious effect of the same kind is produced 

 by the expansion of the circle from change of temperature ; it is 



