1890-91.] Mr E. Sang on NicoVs Polarising Eye-Piece. 
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whence squaring 
( - a 2 sin v 2 + /3 2 cos v 2 ) 2 y 2 (a 2 sin v 2 + /3 2 cosi/ 2 - < Mr ^ 
= a 4 /3 4 cos v 2 sin v 2 ^3a 2 sin v 2 + 3 /3 2 cos v 2 - 
which is an equation of the fifth order, sin v 2 being the unknown 
quantity. 
For the convenience of development put 
a 2 sin i/ 2 = A ; /3 2 cosv 2 = B; ^ = G: 
T 
then the equation becomes 
( - A + B) 2 y 2 (A + B - C) 3 = AB(3A + 3B - 2C) 2 a 2 /? 2 , 
or 
{A 5 + A 4 B - 2A 3 B 2 - 2A 2 B 3 + AB 4 + B 5 } + C{ - 3 A 4 - 9A 3 B - 1 2A 2 B 2 - 9AB 3 - 3B 4 } 
+ C 2 {3A 3 + 9A 2 B + 9AB 2 + 3B 3 } + C 3 { - A 2 - 2AB - B 2 } = 0 . 
Or, again, 
(A + B) 3 (A - B) 2 - 3C(A + B) 2 (A 2 + AB + B 2 ) + 3C 2 (A + B) 3 - C 3 (A + B) 2 = 0 , 
hence as two of the solutions of the equation we have A + B = 0, 
the roots of which are clearly imaginary. The remaining solutions 
belong to the equation 
(A + B)(A - B) 2 - 3C(A 2 + AB + B 2 ) + 3C 2 (A + B) - C 3 - 0 , 
which is only of the third order in reference to the unknown 
quantity sin v 2 . This equation may be put under the form 
sin A 2 (2/3 2 - * 2 ) 2 + sin >^{3a?(/3* - /S 2 e 2 + e 4 ) - y 2 (4 / 3 * - e 4 )} 
+ sin v 2 ^ { 3aV - 3y 2 (/3 4 - € 4 ) + y 4 (4y3 2 - « 2 ) } - ^(y 2 - a 2 ) 3 = 0 . 
Put here sin v 2 = x) 
,2 > 
and we have 
a 3 . e 2 (2/3 2 - € 2 ) 2 + X 2 { 3a 2 (/5 4 - /3 2 e 2 + e 4 ) - y 2 (4£ 4 - € 4 ) } 
+ X{ 3a 4 e 2 - 3a 2 y 2 (/3 2 + e 2 ) + y 4 (4/3 2 - c 2 ) } - (y 2 - a 2 ) 3 = 0 , 
which becomes on substituting the numerical values for a 2 , /3 2 , e 2 , y 2 , 
