402 ON THE MEAN PLACES OF 84 FUNDAMENTAL STARS. [48 



The following is the process to be employed in reducing the above star 

 places from 1800 to 1800 + ?. 



First to the above places for 1800 apply the proper motion for t years. 



Let the resulting Right Ascension and Declination be called a and 8 

 respectively. Take out from the above table the values of 6, z and z' for 

 the year 1800 + *. 



Then if a' and 8' be the Right Ascension and Declination for the year 

 1800 + ?, these quantities will be obtained from the following Equations. 



Assume 



Then 

 and 



cos(a + z) 

 tan rf> = - s-^- 

 tan 8 



tan (a' - z') = -Trra\ tan ( a + z )> 

 sin (<p v) 



s , cos (a' -2') 

 tan 8' = - -7-7 ^ . 

 tan (</> - 0) 



As a check the following formula may be employed, 

 sin (a + z) cos 8 = sin (a' z') cos 8'. 



But as a more severe check, and in order to find still more accurately 

 the places for 1800 + t, we may employ the following. 



Let a + z = A, a!-z' = A'. 



Then 



sin %(A'-A) = sin $(A' + A) tan $ (8' + 8) tan $ 0, 



, /5 , 5X cos %(A' + A) 

 tan i (8' - 8) = - *-,-.-. - -^ tan A 0. 

 cos (A' -A) 



The differences A' A and 8' 8 may be more accurately found from 

 the logarithmic tables by these formula than A' and 8' themselves can be 

 by the formulae given before. 



The above was the process followed by Mr Farley, except that he calcu- 

 lated the values of 0, z and z' for each 4th year, differenced the results 

 and interpolated the places for every year. 



[Here follow the star places thus found for every year from 1830 to 1870.] 



