o/TresnePs Optical Theory of Crystals. 463 



tion is expressed in terms of the <es which it makes with the 

 optic axes, 



viz. v* — b 2 = a 2 — c % . cos e / . cos e /t . 



In the 12th proposition e, e /y are separately expressed in 

 terms of i, i„. 



In the Appendix I have given the polar or rather radio- 

 angular equation to the wave surface, from which the celebrated 

 proposition of the ray flows as an immediate consequence. 



Proposition 1. 



Let Ix -f my -f n z = . . . (a) be the = n to a given front 

 to determine the lines of vibration therein. 



It is clear that if x / y / z be any point in one of these lines, the 

 force acting on a particle placed there when resolved into the 

 plane must tend to the centre. Consequently the line of force 

 at x y z must meet the perpendicular drawn upon the front 

 from the origin. Now the = n to this perpendicular is 



I m n v 



And the forces acting at x y z are a 2 x, W y, c q z parallel to 

 x y z, — so that the = n to the line of force is 



X-g _Y-y_Z-« 



a*x "7 b 2 y ~~ c*z ' l ; 



f% X - a*x Y = (& - a f ) a: # (3.) 

 from (2) we obtain J c 2 zY - b 2 yZ = (> - b*) y z (4.) 



La*a?Z - c 2 *X = (a 9 -c*)sa; (5.) 



Hence (Z> 3 — a*)xyn + (c 2 - b 2 )yzl + (F-c^zxm 



= b i y<JZ-nX)+c' 2 z(mX- I Y) + a*x(n Y-m Z) 

 but by & z = ns (1) Z Z - n X = m X-Z Y=0 it Y-ro Z =0 



... (ja _ „•) JL + ( c * - #) i. + (a 2_ c «) J5. = o . (b) 



Also we have nx + Ix + my = , («.) 



... (£* _ fl «) n * + (<J - £«) /2 + n / . (* 2 - £* . i+6*-a».£-^ 



= a 2 — c 2 . ;»* 



2 



(^-^(-j) + ^7-{c 2 -6 2 .Z 2 + 6*-a 2 .n 2 -.a 8 -c 2 .w 9 }^ 



