g/TresnePs Optical Theory of Crystals. 467 



or, if we please, 



/ (q«q»-l)(q»^ fp^TlHP-*) 



V (a 2 - 6 2 ) (a 2 - c 2 ) ' T V (£ 2 - a 2 ) (6* - c 2 ) y 



• V (c* - a 2 ) (<* - 6 2 ) 



and eliminating by differentiation (w) and (0) we obtain 



a 9 # 9 ft 2 v 2 c 2 # 2 



a 2 -^ 2 -^ 2 ** 2 ) + P-aP + tf + x* + c»~**+y*+» 9 = V 



Proposition 7. 



To find when », = v w . 



By Prop. 4, 



* y fa 9 - « 2 ) _ y.to'-fr) = *, fa* - * 3 ) 

 *«(V-«") *>,/-**) <W-*) ' " 



Hence when & = v it we have, generally speaking, 



(«•) 



*« y» */, * 



Now ^ *»J, + y, y„ •+ *, «j| = 



... j. a _j_ yj. _j_ ^ -2 wou l ( ] s- o, which is absurd. 

 The only case therefore when v, can = v n is when one of 

 those terms of = n (0) becomes — : thus suppose y = b, then 



we have — I = ' = — , and we can no longer infer — ' = -^. 



Let now (eo, <$i \(/,) (co„ <p tl ^ n ) be the two systems of values 

 which w, $, \|/ assume when v, = u /y = &, then applying the 

 = n of Prop. (5) we have 



cos ., = y ^r^ cos «„ = ^ ^— ^ 



cos $, = cos $„ = 



. /&» -c 2 J / 6 2 - c 2 



so that (6) must correspond to the mean axis. 



Proposition 8. 



i, % H being the < es made by the front with the optic planes 



to find t t i u in terms of v t v lt . 



302 



