290 APPENDIX. 



which can be put in the form 



x = r sin a sin ( A -j- u) 



(2) y = r sin b sin (B -\- u} 



s = r sin c sin ( -\- u) 



or 



x = r sin a sin A cos t -j- r sin a cos ^4 sin u 



(3) ^ r sin & sin 5 cos u -\- r sin b cos .B sin M 

 z =. r sin c sin (7 cos M -(- r sin c cos C&quot; sin M 



equations (3), compared with (1) give 



sin a sin 4 = cos Q, sin a cos A = sin & cos i 



(4) sin b sin 5 = sin Q, cos e sin i cos B = cos S cos i cos e sin? sine 

 sin c sin (7 sin S sin e sin c cos C = cos 8 cos ? sin e -j- sin * cos e . 



By introducing the auxiliary angle E 



tan t 



we shall find 



cotan A = tan Q cos 



tun Q cos .& cos 



cotan (7= -- ^+l)_ 

 tan 2 &amp;lt; * -asm s 



_ cos Q _ sin a cos 



Dill t* - . ; - ~r - 



; - ~r - -- 



sm A cos ^. 



j _ si&quot; S2 cos e cos Q cos i cos sin f sin e 

 bill c&amp;gt; . - . y. z^i - - 

 sin B cos .5 



_ sin Q fin e cos Q cos i sin 4- sin f coss 

 sin c - : -~ --- - . 

 sin U cos C 



sin a, sin i, sine are always positive, and the quadrants in which A, B, C are to 

 be taken, can be decided by means of equations (4). 



The following relations between these constants, easily deducible from the 

 foregoing, are added, and may be used as checks : 



tan ; _ ain5sincsin(O B) 

 sin a sin A 



