SECT. 1.] THREE COMPLETE OBSERVATIONS. 193 



thenco, and from 



f\ o/w + w&quot; i\ /s j i Ksmtf 

 Q = 2 ( ~- -- 1 1 r 3 and r = 



\ n smz 



is obtained, 



Qsinz* i P+l . , 



sm * + 2*** = J P +^ sm (* ) or &amp;gt; 



Qsm*z /7-P+l \ 



^h^ = \ b P^ ~ C S / Sm ^ ~ &amp;lt;T ) ~~ Sm a COS ( S ) 



Putting, therefore, for the sake of brevity, 



and introducing the auxiliary angle to such that 



sin a 



tan to = - , l 



cos a , 



we have the equation 



IV. c Q sin w sin 4 g =: sin (2 w 0), 



from which we must get the unknown quantity z. That the angle w may be 

 computed more conveniently, it will be expedient to present the preceding for 

 mula for tan co thus : 



Whence, putting, 

 5 



a 



ri K-I cos JT 



[ ] x~ii = 



COSff 



[16] 



COSff 



we shall have for the determination of to the very simple formula, 



tan to = 



P+d 



We consider as the fourth step the computation of the quantities a, b, c, d, e, 



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