APPENDIX. 301 



For C, article 100, it is only necessary to write g in place of x in the pre 

 ceding formulas. 



A may be computed more conveniently by the following formula : 



a/ 1.5 1.3.5.7 ,, 1.3.5.5.7.9 



A=(l-x] H 1 +279^ + 2^ 



. 4.6. 9. 11. 



142. 



PROF. ENCKE, on the 13th of January, 1848, read a paper before the Royal 

 Academy of Sciences at Berlin, entitled Ueber den Ausnahmefall einer doppetten 

 Bahnbestimmung ans denselben drei geocentrischen Oertern, in which he entered into a 

 full discussion of the origin of the ambiguous case here mentioned, and the 

 manner in which it is to be explained. The following paragraphs, containing 

 useful instructions to the practical computer, embody the results of his in 

 vestigation : 



By putting 



m = c Q sin w 

 = (oi-f a), 



Equation IV., 141, becomes, for r ^&amp;gt;Hf 



m sin 4 s = sin (0 q) 

 and for / &amp;lt;^ R 



m sin 4 s = sin (z -)- q) 

 m is always positive. 



The number and the limits of the roots of this equation may be found by 

 examining both forms. 



Take the first form, and consider the curves, the equations of which are 



y m sin 4 z, y = sin (z q) 



y and y being ordinates, and z abscissas. 

 The first differential coefficients are 



dy . dt/ , . 



^| = 4 m sin 4 z cos z, ~- = cos (z q), 



