g^FresnePs Optical Theory of Crystals. 465 



A % y,/ 



±__x, a 2 -b 2 . y? - c 2 — a 2 . z* 

 ' ™ m Vi ' b*^~? . z 2 - a % - b 2 • *? 



*l a * • (** + y? +•*') - ( gg ** + ^ g + c * ** ) 



" y, " b* . (*« + y 2 + */) - (««*« + Vy? + c> z*) 

 If now we make x, 9 + yf + zf — \ 



*efjr b 2 yf + c*z? = v 2 



,\ a*—vf x t . x + b 2 — vj* . y, . y + c 2 — v, z, . z — is the 

 = * required. 



Proposition 4. 



— , — having each only one value, shows that only one 



front corresponds to the given line of vibration. Let x u y n *~ v n 

 correspond to x t y { z f v t for the conjugate line of vibration, 

 then the = n to the front may be expressed likewise by 



so that 



v* x u x + b 2 - v 2 .y,,y + c 2 — v 2 z u . z = o, 



{p-vtfixi i&-v u *)y H (c 2 -V)V 



Proposition 5. 



To find co, cp, rj/, the < es made by the front with the planes 

 of elasticity in terms of v t v u . 



By the last proposition 

 , cos .). - {a 2 -v 2 Tx* . 



Now, by proposition (2), £2$ = J^ 2 = ~^- t 



.'. (COS CO 2 ) 



(«• - tyQ (a 2 - tyQ (c» - 6 2 ) 



TAW Sm«. Vol. 1 1. No. 69. Not. 1837. 3 O 



