78 RELATIONS PERTAINING SIMPLY [BoOK 1. 



This ambiguity in the determination of a by the second equation is removed by 

 this consideration, that cos a and cos I must have the same sign. 



This method is less expeditious, if, besides a and d, E also is required : the most 

 convenient formula for determining this angle will then be 



sin s cos sin e cos I 



- ;- ^r. 



cos b cos o 



But E cannot be correctly computed by this formula when + cos E differs but 

 little from unity ; moreover, the ambiguity remains whether E should be taken 

 between and 180, or between 180 and 360. The inconvenience is rarely 

 of any importance, particularly, since extreme precision in the value of E is not 

 required for computing differential ratios ; but the ambiguity is easily removed 

 by the help of the equation 



cos b cos S sin E = cos sin b sin d, 



which shows that E must be taken between and 180, or between 180 and 

 360, according as cose is greater or less than sin b sind : this test is evidently not 

 necessary when either one of the angles b, d, does not exceed the limit 66 32 ; 

 for in that case sin E is always positive. Finally, the same equation, in the case 

 above pointed out, can be applied to the more exact determination of E, if it 

 appears worth while. 



68. 



The solution of the inverse problem, that is, the determination of the longi 

 tude and latitude from the right ascension and declination, is based upon the same 

 spherical triangle ; the formulas, therefore, above given, will be adapted to this 

 purpose by the mere interchange of b with d, and of I with . It will not be 

 unacceptable to add these formulas also, on account of their frequent use : 



According to the method of article 66, we have, 



sin (45 - - H) sin } (E 1} = cos (45 -f } a) sin (45 - - i (e -f &amp;lt;?)) 

 sin (45 } b) cos l(El) = sin (45 -f I a) cos (45 } (e 

 cos (45 * b) sin } (E+ 1) = sin (45 + } ) sin (45- - } (e 

 cos (45 j b) cos } (E -f 1) = cos (45 + } ) cos (45 i (e -f d)) . 



