SECT. 2.] TO POSITION IN SPACE. 93 



The formulas TV. and VIII. already appear in the most convenient form for cal 

 culation ; but the formulas L, HI., V., are reduced to a more elegant form by 

 obvious substitutions, as 



. cos w tan 



di/ 



V.* (-:-J = cos(Z Z)sint&amp;gt; = , cos (L 1) sin b cos b. 



Finally, the remaining formulas II., VI., VII., are changed into a more simple form 

 by the introduction of certain auxiliary angles : which may be most conveniently 

 done in the following manner. The auxiliary angles M, N, may be determined 

 by means of the formulas 



tan M = &quot; , tan N= sin w tan i = tan M cos w sin i. 



COS I 



Then at the same tune we have 



cos 2 M 14- tan 2 N cos 2 i -4- sin 2 ta sin 2 a 



, ^ _ I _ n -. _ I _ i f\ /&quot;iO /|J * 



cos 2 .AT &quot;~14-tan 2 J^~ cos 2 f -f tan 2 w 



now, since the doubt remaining in the determination of M, N, by their tangents, 

 may be settled at pleasure, it is evident that this can be done so that we may 



have 



cos M , 



and thence 



sin 



vj=-. 



sin M 



These steps being taken, the formulas IT., VI., VII, are transformed into the fol 

 lowing : 



TT * __ 



~ 



m cos 



\di ~ 4 sin M 



VI* (^) = -^-(coscu smicos(M w)cos(JV 



&quot;irrr * (^^\ __ r s n M cos * cos ^ ^ 

 \d i) ~ A cos N 



