280 LORD M'LAREN ON DIFFERENTIAL REFRACTION. 



The analytical investigation of these corrections leads to the following- 

 expressions : — 



If we call tt and -k the true and apparent position angles ; A' and A, the 

 true and apparent distances of the two stars ; £, the mean zenith distance of 

 the field of view ; n, the parallactic angle ; and k, the co-efficient of refraction, 



we have 



•a-' = 7r — k tan 2 £ sin {-k — >i) cos (ir— rj). 

 A' = A + K A [1 + tan 2 f cos 2 (« - >?)]. 



In these expressions all the variable quantities are given directly by the 

 readings, excepting tan 2 £ and 17, the parallactic angle. Now, the last men- 

 tioned quantities are functions of the latitude, declination, and hour angle. 

 They can therefore be tabulated for a given latitude, with the arguments, 

 declination, and hour angle. The present tables give for the parallel 55° 56' 

 (which passes through Edinburgh), and also for 57° 30' the quantities log tan 2 £ 

 and n for each ten minutes of hour angle, and for each interval of two degrees 

 of declination from 40° north to 90°. The tables include the entire circumpolar 

 region of the heavens visible from the respective latitudes, and one or other of 

 them may be used for observations taken in any part of Scotland, without 

 sensible error. Where great accuracy is desired, a table of differences 

 applicable to the particular observatory may be obtained by interpolating 

 between the two printed tables. 



The computations for the two tables were made in the following manner : — 

 Calling cf> the latitude of the place of observation ; II, the polar distance corre- 

 sponding to the interval of declination ; and r the hour angle — the quantities 

 £ and n are to be obtained by solving the spherical triangle, whose vertices are 

 the pole, the zenith, and the star ; whose sides are polar distance, zenith 

 distance, and the co-latitude ; and whose angles are hour angle, azimuth and 

 parallactic angle. 



To adapt the solution to logarithmic computation, the auxiliary angles 

 M and N were computed for each 10 minutes of hour angle by the formulae 



Sin M = cos sin t. 

 Tan N = cotan <p cos r. 



The resulting values of N and log cos M were tabulated, and the final compu- 

 tations were made by the formulae 



Cos f = cos M cos (IT - N). 

 Cos t] = cotan £ tan (II — N). 



The quantities log tan 2 £ and n were directly computed for each alternate 

 column of the tables. The intermediate columns were obtained by interpola- 

 tion, checked by independent computation of a sufficient number of tabular 

 places to ensure substantial accuracy in the last decimal place. 



