CHAP. XIII.] OBSERVATIONS FOR ERROR 319 



In working the rigorous expression, an approximate value 

 for a is first obtained, disregarding the term Cos. a, and then 

 rework, using the vahie of a thus found. 



Notes on the Computation. 



The constant logarithm 4-138339 is an abridgment of Chau- 

 venet's formula, by adopting it to circular measure, and is 

 correct so long as the equation of equal altitude is small ; 

 certainly up to four minutes no error is introduced by its use. 



The cosine of a with an equation of equal altitude under 

 four minutes changes so slowly that it is practically the same 

 for each set unless separated by a very long interval. 



The R. A. M. S. must be found for the Greenwich date 

 corresponding to the clu'onometer middle time. In cases of 

 a small equation of equal altitude and changing slowly, this 

 is the same for each set ; but where it is large, and therefore 

 changing rapidly, as in the second example, the R. A. M. S. 

 must be corrected for every set. But this merely involves 

 applying the acceleration due to the difference of the chrono- 

 meter middle time of each successive set to the R. A. M. S. 

 for the first set. 



A more or less accurate knowledge of the G. M. T. is there- 

 fore necessary, but tliis is inherent to the use of stars for time 

 under all circumstances, and a second approximation must 

 be made if the error on local time has been assumed more than 

 three or four seconds in error. 



It is worthy of remark that tan c^xtan 8x4-138339 and 

 tan Zxtan gx 4-138339 form absolutely constant logarithms 

 for the same stars, at the same place, for any night in the 

 year, and it is only necessary to add the logarithm of | E. T. 

 to each to obtain the equation of equal altitude corresponding 

 to that ^ E. T. 



The equation of equal altitude is always so large as to 

 exclude all possibility of doubt as to which way to apply it, 

 but the investigation gives the rule. Two examples are 

 given ; the first illustrates the cases of two stars differing 

 by 1° 40' in declination, and the second where they differ 

 by 8° 25'. 



In the first instance the equation of equal altitudes hardly 



