'206 PROCEEDINGS OF THE AMERICAN ACADEMY 



to be identical with those of the biquadratic ; and consequently 11 is 

 determined by the equation 



tan [tan M + cot,.] + 2 *" ^ M sin (0 - ,.) cos (0 + M ) = C-l 



L r ' rj ' cos 3 /3 cos 2 2 /i c 2 cos 2 /3' 



or 



c 2 sin 2 c 2 sin 2 /. [sin 2 /x — sin 2 0] 2 



sin 2 /x 1 — sin 2 2 ti ' 



or 



^3 



sin 3 2 it -f (c 2 — 1) sin 2 /i — c 2 sin 2 = 0. 



That this cubic will always give at least one real value for p, is evi- 

 dent on making in the left-hand member sin 2 ti successively equal to 



— 1, 0, and -j- 1 ; the results obtained are 



— c 2 (1 -f- sin 2 j8), always negative ; 



— c 2 sin 2 0, negative or positive, according to the sign of sin 2 ; 

 -(- c 2 (1 — sin 2 0), always positive. 



Moreover, it is plain that there is one real value of /., which makes 

 sin 2 n and sin 2 have like signs ; this value we shall adopt. 

 Making, according as e 2 is greater or less than unity, 



c 2 = sec 2 , y, or c 2 = cos 2 y ', 



the above cubic is solved by these formulae (see Chauveuet's Trigo- 

 nometry, p. 96), it being necessary to make three different cases. 



tan <f> = 



Case I. 

 2 sin 2 y tan y 

 VH sin 2 ' 



tan yf/ = tan ~, 



2 

 sin 2 a = -7= tan y cot 2 \Jr. 

 ^3 ' r 



Case n. 



2 sin 7' tan 2 y 

 sin <£ = — 7=^ '-, 



V 27 sin 2/3 



tan y = tan —, 



. -, 2 . 



sin z u = — — sin v cosec 2 sir. 

 ^3 r 



