316 FRESNEL ON DOUBLE KEFBACTION. 



We may put the polar equation of the ellipsoid under the form 

 and substituting for («) and (/3) their values, we have 



+ {t-g)[p'{t-h) + qHt-f)Y=0; 

 or 



(t-f){t-h)lp'{t-h)+q^{f-f)-] 



+ {t-ff)[pHi-h)+rf{t-f)V = 0; 

 or lastly, suppressing the common factor^/ {i — h)+q'^ (^— /)? 



{t-f) {t-h) +;/ {t-g) {t-h)+rf [t-f) [t-y) = 0, 

 an equation of the second degree, which ought to give at the 

 same time the maximum and minimum values of {t), that is to 

 say, the two values of {t) which correspond to those of the semi- 

 axes of the elliptical section. 



We may divide this equation by {p'), and put it under the 

 form 



[t-f) {t-h).j, + [t-g) [t-h)+t ^t-f) {t-g) = 0. 



1 2 



And substituting for -5 and ^ the values which we have above 



found in functions of the angles (m) and (w), we arrive, after 

 several reductions, at the equation 



f- — t. [/+/« — (/—/«) cos w. cos m] +/A + — (cos'-^w + cos^ot)(/— A)^ 

 — — cos n . cos m (/'-^ — /<^) = ; 



whence we obtain 



t = — . {f + h) — Tr[f — h) cos n . cos m 



+ — {f — h) \/ 1 + cos^ n cos^ m — cos^ n — cos^ m, 

 or 

 t =. — (/+ A) — — if—h) cosw cosm + -^{f — h) sin w . sin »i* : 



* The two values of (t), which give the quotients of unity divided success- 

 ively by the squares of the velocities of the ordinary and of the extraordinary 

 ray, may be put under the following form : — 



i= i (/+ h) - i (/- h) cos {m + «), 

 ^"•^ ^ = 1 (/ + //) -~~{f- h) cos {m - u). 



I 



