268 Dr. L. Trenchard More on the Coincidence of 



Returning to the general equation (a), it may for more 

 convenient manipulation be put into the form of a quadric 

 quadric, 



; 4 (\^ 4 + X 2 ,r 2 + X 3 ) + M(w? + ^ + fh) + ( V 4 + v^ + v 3 ) = : 

 where 

 Xl =(a 2 - C % 



X 3 =c 4 (a 4 -&V) 2 , 



/* 2 = {2a W (a 6 + c 6 ) - aV (a 2 + fc 2 ) (^ + e 2 ) - 2a V (a 2 - //) (?, 2 - c 2 ) 

 /* 8 = - a W(a 4 - JA- q ) (a* ~ c 2 ) = - V Vs, 



Vl = a 4 (a 2 ^-c 4 ) 2 , 



v 2 = - 2aW (a 9 - - c 2 ) (a 2 J 2 - c 4 ) =-2 Vv^, 



v 3 = aW(« 2 -c 2 ) 2 . 



The asymptotes parallel to the axes are the roots of 



X 1 x 4 + \ 2 so 2 + X s = 0, 



V 4 + 2 f i 1 z 2 + v 1 =0. 

 Those for a negative uniaxal crystal reduce to 



a 9 - + h* 



x=+b 



z= + 



■a 2 -IP 

 ab 



a + b 



The locns of a crystal intersects the axes in the points 

 which are the roots of 



v 1 z 4 + v 2 x 2 + v 3 =0, 



X 3 z i + 2 f i 3 z 2 + v 3 =0. 



Inspection of these coefficients shows that botli equations 

 are perfect squares. Substituting their values we have 



a 1 — c 2 



r 2_7,2 2_^ L 



„o a 2 — c 2 



