280 APPENDIX. 



e&quot; being used to denote c expressed in seconds, then we have 



or 



M M = E s e&quot; (sin E sin e ) 

 = (E t) (1 ecoss), 



if E e is regarded as a small quantity of the first order, and quantities of 

 the second order are neglected for the present : so that the correction of K is 



M M 1 



1 e cos s 



and a new approximate value of is 



. M M 



1 e cos s 



with which we may proceed in the same manner until the true value is obtained. 

 It is almost always unnecessary to repeat the calculation of 1 e cos e. Gener 

 ally, if the first is not too far from the truth, the first computed value of 

 1 e cos may be retained in all the trials. 



This process is identical with that of article 11, for X is nothing more than 



. _ d log sin E _ cos E 



&amp;lt;i i: ~ shTlr 



if we neglect the modulus of BIUGGS S system of logarithms, which would subse 

 quently disappear of itself, and 



d (e&quot; sin E) 



therefore, 



n ). i i 

 and 



n M M 



.- ~, 



p + /I 1 e cos E &quot; 



and the double sign is to be used in such a way that &amp;gt;. shall always have the same 

 sign as cos E. In the first approximations when the value of differs so much 

 from E that the differences of the logarithms are uncertain, the method of this 

 note will be found most convenient. But when it is desired to insure perfect 

 agreement to the last decimal place, that of article 11 may be used with 

 advantage. 



