294 APPENDIX. 



59. 



If r, b, and / denote the radius vector, the heliocentric latitude and longitude 

 of any planet, the rectangular coordinates referred to three axes, of which 

 that of x is directed towards the vernal equinox, that of 0, parallel to the earth s 

 axis, and that of y, 90 of right ascension in advance of x, will be as in case II. 



x = r cos I cos I 



y = r cos b sin I cos t r sin b sin e 

 z =. r cos b sin e sin l-\- r sin b cos 

 and by putting 



cos u = cos b cos I 



sin 5 sin ? cos b 

 Sin U = -.-- = - 



sin cos o 



. tan b 



tan 6 = -^ , 



fin I 



they assume the following forms convenient for computation : 



x = r cos u 



y := r sin M cos (6 -f- e) 



2 =z r sin ?&amp;lt; sin (0 -j- c) . 



74. 



The following are the solutions and examples from the Monatltche Correspon 

 dent referred to in this article, adopting the notation of article 74, and using I! 

 to denote the longitude of the Sun. 



Given, &, L , I, b, i, R, to find u, r, 4 , and the auxiliary angles A, JB, C, etc. 



L 



9 sin (L l) tan t _ cos -B sin 6 tan (/, Q ) _ 



^i , F / , , tcin JL&amp;gt; -- 



, F / , , c J . ^ -^ j ir -- i 



oos (/y Q ) sm (B -\- b) cos % 



3. ^ ^- 8 &amp;gt; !* = tan C -^2P r ?- ~^ = tan u 



B\n(L Qtani sm(0-\-L Q ) cos i 



cos (L Q) tan 4 _ sin .P tan (Z, Q ) co*(L l) _ 



- FT/ - y\ *i - Lclll X/ 7&quot;VT~i - */ - iu -- - &quot; - will (( 



cos (U I) tan i sin (D -f- L /) cos i 



