OF ARTS AND SCIENCES : FEBRUARY 8, 18T0. 207 



Case III. 



sin 3 0= V^si"^ , 

 2 sin y' tan 2 y' 



2 

 sin 2j* = — sin y' sin (0 ± 60°). 



V o 



When $ is impossible in Case II., the formulas of Case III. must 

 be used; and the upper or lower member of the double sign in the 

 second equation must be taken according as sin 2 /3 is positive or nega- 

 tive ; in order that sin 2 /x may have the same sign with sin 2 fi. All 

 the auxiliary angles (f>, \^, and fi may be taken between the limits 

 ± 90°. Since sin 2 fi sin 2 /x is always positive, tan fi tan fi and tan 

 fi cot fi are so likewise, since they are respectively equivalent to 



sin 2 fi sin 2 /u sin 2 /3 sin 2 /a 



j 2~o 2 and - — . . „ . 



4 cos' fi cos 11 /x 4 cos- fi sin- /x 



Let us take two auxiliary angles 6 and ff, determined by the equa- 

 tions 



. _ tan? fi tan* /x cos fi cos 2 /x 



Sin - 8 -. t-t — : r , 



sin 2 ff = - 

 or by the equations 



sin 2 6 = q: 



sin /x cos (fi -4- fi) 



tan* fi cot* ii cos /3 cos 2 /x 

 cos /x sin (fi — li) 



cos 2 fj. /sin 2 /3 



1 + /0 V si 



cos 03 + /x) y sin 2 /x' 



. . /1 , cos 2 u /sin 2 fi 



Sm 2 ' = T sin 03 - M ) V smT^' 



where the upper or lower of the signs must be taken according as 



cos fi . . . cos fi . . , 



- in the first and in the second are positive or negative ; 



sin n cos /x ° 



and 2 5 and 2 ff may also be taken within the limits ± 90°. The four 

 values of x or tan a are then 



tan a = tan* /3 tan' ^ tan #, 

 tan a = tan* /3 tan* /x cot 0, 

 tan o- = tan* fi cot* /x tan ff, 

 tan o- = tan* fi cot* /i cot ff. 



If the value of sin 2 or of sin 2 ^ does not fall within the limits 



