OF CAMBRIDGE OBSERVATORY. ' ' 27T 



to take sines of even multiples, or cosines of odd multiples, because 

 when 180° — Z.D. is substituted for Z. D., the result is equal in 

 magnitude but opposite in sign ; and therefore when the two are 

 added together, (as they are in finding the zenith point from each 

 star), no trace of these terms would remain. Thus there may be 

 sensible flexure in the circle which cannot be discovered from ob- 

 servation by reflexion. The sines of odd multiples, and the cosines 

 of even ones, (all which may be expressed in finite series of powers 

 of sin Z.D.), will produce the same values with the same signs for 

 180° — Z.D. as for Z. D., and these will affect the zenith point. 

 Thus it appears that the terms which aff*ect the zenith point are 

 the same for a direct observation and for the corresponding observation 

 by reflexion, and it is this which justifies us in applying half the 

 discordance to each. It appears also that when Z. D. = 90°, the 

 function is maximum or minimum, and hence the curve in the 

 graphical process above described must there be parallel to the line of 

 abscissEB. 



The tables of corrections being thus formed, I now considered 

 myself entitled to apply them to the reduced r^ults of all the 

 observations, whether there were corresponding observations of the 

 opposite kind or not. ' >»/ ^ i - 



The principal steps of the succeeding process may be gathered 

 from the subjoined table. The first column contains the name of 

 the star, its position with regard to the pole, (the lower transit being 

 marked by S.P.), and the method of observing it (the letters D and 

 R being always used for direct and reflected vision). Here it is to 

 be observed that a star above the pole and the same star below the 

 pole are reduced as separate stars, which is necessary, because the 

 observations have been reduced with an assumed co-latitude, or an 

 assumed place of the pole, the error in which assumption can be 

 found only by comparing the separate results for the same star above 

 and below. The second column contains the number of observations. 

 The third contains its mean N.P.D. for Jan. 1, 1833, as found from 

 the mean of all the results in each position and mode of observation. 

 Vol. V. Part II. Nn 



