27 



expressed in constants and sines or cosines of the multiples of R, 

 the constants disappear in taking the difference of the arcs mn'B 

 and mn', and the rest after integration assume the above form in 

 which the part which is not proportional to is multiplied by a 

 factor of the form 



..(ix«..) 



sin. Sjr 



sin. I— a. ^ sin. — x 



It therefore vanishes except when s -j- m is an integer, being then 

 ~, and having the value ±n. Two microscopes therefore correct 



for excentricity or any other error varying according to the same 

 law, as also for all which are as odd powers of the sine or cosine ; 

 three are still better, failing only where the number expressing the 

 order of the error is divisible by three. 



In general any error which is reducible to a series of these 

 functions, or rather to the difference of two such, can be corrected 

 by a certain number of equidistant readings, (a) 



In Dr. Brinkley's circle the excentricity is not totally corrected, 

 for a = ^, and w = 3, one microscope being wanting to complete 

 the circle. The uncorrected error is 



^e.sin.(|). sin (|-j) 



3 



which is a maximum when the altitude observed = g, and be- 

 comes then -rj-Trsfw^— /n-^-'^-'^'^- ^ ■ 



— <r. sin.«(J{). 



But there does not seem to exist any excentricity in this in- 

 strument. 



It may be remarked that no error is committed by assuming = ze, 



e2 



