276 PROFESSOR AIRY ON THE LATITUDE 



selected for the three first positions of the telescope all the un- 

 exceptionable corresponding observations D and R. (The stormy 

 weather of December made it impracticable to observe low stars by 

 reflexion). In each case of a double observation, the difference of 

 the results D and R would be double the difference between the 

 zenith point as found from that star, and the zenith point adopted 

 in the reductions. The mean of the differences of all the correspond- 

 ing results D and R, would therefore be double the mean of all 

 the differences between the zenith points found from the particular 

 star, and the zenith points found from all by a tolerably uniform 

 system : and thus it might be considered as double the difference 

 between the zenith point found without error of observation from 

 that star, and a certain imaginary well defined point. These values 

 for all the stars, and for each position of the telescope, were arranged 

 in tables (for which, as well as for some other numerical values, I 

 must refer to the Cambridge Observations, Vol. VI.) 



The next step was, to connect these, approximately at least, by a 

 law. I soon found that to attempt this by calculation was almost hope- 

 less. Combinations of constants, sin Z.D., sin Z.D. cos^ Z.D., cos2Z.D., 

 were tried in vain. I therefore adopted a graphical method similar 

 to that used by Sir John Herschel, in the reduction of his sweeps, 

 and described by him in the Phil. Trans. 1833. Taking the line of 

 abscissae for zenith distances, and the ordinates to represent the mean 

 of the differences above mentioned, I made a curve to pass among 

 the points so determined, as well as I could, giving to each point 

 an importance depending on the number of observations. From this 

 curve I measured off" the ordinates for every 10° of zenith distances; 

 half of this quantity I considered to be the correction to the ob- 

 served zenith distance, to be applied with different signs to the 

 direct and the reflected observation. The only respect in which 

 theoretical consideration may be said to have assisted me is the 

 following. Since the error in the relation between the position of the 

 telescope and the reading of the circle, to which the discordance 

 must be due, is periodical and never infinite, it may be expressed by 

 sines and cosines of the Z. D. and its multiples. Now it is useless 



