368 



suppose the clock equally audible in every position of the te- 

 lescope, and the observer able to observe in all with equal 

 convenience, for personal discomfort will interfere with the 

 attention). That arising from uncertainty in estimating the 

 star's place will vary inversely as the cosine of its declination. 

 " But there is yet another, arising from the actual dis- 

 placement of the star's image, by irregular changes in the re- 

 fractive density of the atmosphere; the effect will, as the 

 preceding, be inversely as the cosine of declination, but also 

 directly as the magnitude of these changes. This depends, 

 in the first place, on the heterogeneity of the air as to heat 

 and moisture ; and in the second, on the quantity of dis- 

 turbed medium through which the line of sight passes. The 

 former scarcely admits of expression in terms of our present 

 meteorological data, and we must be content to assume for it 

 an average value. In respect to the other, as the disturbance 

 takes place chiefly within a small distance of the earth's sur- 

 face, it will easily be seen that its amount is as the secant of 

 zenith distance. If then we denote by u the probable error 

 of the ear, by y that of the eye, and by z the atmospheric 

 tremor at the zenith, we have, by the theory of these errors, 

 for a star whose declination is 8, 



cos 3 . § cos 2 S . COS 2 (I - S) 



It is evident that z admits of a minimum in respect of § : let x 



z 

 be the tangent of / - 8, A that of latitude, and r = -, differen- 



... . y 



tiatmg and equating to 0, we derive 



„ 2 , 2>- 2 + 1 r 2 + 1 



x A + T x 3 + — = 0. 



A \r- r 2 



" If then we select three stars, properly differing in zenith 

 distance, we can determine the three errors u, y, and z. We 

 find £ in the usual way, by comparing each wire of a set of n 



