SECT. 2.] 



TO POSITION IN SPACE. 



77 



sin (45 - - id) sin } (E -f a) = sin (45 + H) sin (45 } (e + 5)) 

 sin (45 J d) cos } (^ + a) = 6os (45 -f /) cos (45 J (e )) 

 cos (45 * J) sin l(Ea) = cos (45 + H) sin (45 } (E 5)) 

 cos (45 } 8) cos (, a) = sin (45 -j- } /) cos (45 } (e -f 5)) 



The first two equations will give i(^-)-a) and sin (45 - -i&amp;lt;?); the last two, 

 l(Ett) and cos (45 i d) ; from J (^ + ) and i(j? o) will be had a, and, 

 at the same time, E ; from sin (45 k d] or cos (45 i $), the agreement of 

 which will serve for proving the calculation, will be determined 45 id, and 

 hence S. The determination of the angles $ (E -\- &amp;lt;*}, $ (E ) by means of 

 their tangents is not subject to ambiguity, because both the sine and cosine of the 

 angle 45 J 8 must be positive. 



The differentials of the quantities a, 8, from the changes of I, b, are found 

 according to known principles to be, 



, sin -Scos 5 n 7 cos E -, -, 



&a = - . til- --- ^-db 



COS COS 



d d = cos E cosb d /-(- sinJEdb. 



67. 



Another method is required of solving the problem of the preceding article 

 from the equations 



cos e sin I = sin e tan b -\- cos I tan a 



sin d = cos sin b -|- sin cos b sin / 

 cos b cos ^ = cos a cos d . 



The auxiliary angle & is determined by the equation 



. tan b 



tan$= -T-T, 



sm/ 



and we shall have 



cos (s -4- 6) tan / 

 tan a= 



cos 

 tan d = sin a tan (e -f- d), 



to which equations may be added, to test the calculation, 



cos b cos I t, cos (e -4- 0) cos 5 sin J 



coso=- ,orcoso = s 1 -~-. - . 



cos a 



-~-. 

 cos sm a 



