300 APPENDIX. 



whence 



dx = -dr -[-# cotan (A -|-) d v -\- x cotan (vl-f-) dn 



[x cotan (A -}- M) -\-y cos e -(- sin e] &amp;lt;/ & -|- r sin M cos a rfz 



dy = y -dr -\- y cotan (.Z? -f- w) e?p -(-y cotan (i? -(- M) J n 



\_y cotan (B -f- M) a; cos e] d & ~f- r sin u cos i ? 



ds = -dr s cotan ( C -j-w) rfw-f- 2 cotan (O-\-u) dn 



[s cotan ( C-\- u] x sin e] d Q, -f- f sin M cos c di. 



These formulas, as well as those of 56 may be found in a small treatise 

 Ueber die Differentialformeln far Cometem-Balmen, etc., by G. D. E. WEYER, (Berlin, 

 1852). They are from BESSEL S Abhandlung liber den Olbers schen Cometen. 



90. 



GAUSS, in the Berliner Astronomisches Jahrbuch for 1814, p. 256, has given an 

 other method of computing , and also C of article 100. It is as follows : - 

 We have 



c_ 5. 10 



~ + = 



This fraction, by substituting for X the series of article 90, is readily trans 

 formed into 



f- 8 &amp;lt;^(-\ I 2 - 8 | 3.8.10^ , 4.8.10.12^ , 5.8.10.12.14 4 



los^v 1 ~9~ &quot; 9.11 ^ 9.H.13 : 9.nm7ur* 



Therefore, if we put 



* 1 I 2 O lO.O.ll/oi , 



^ == l + __^ + ___^ + etc, 

 we shall have 



by means of which 5 can always be found easily and accurately. 



