TO FRESNEL'S WAVE SURFACE. 8^ 



{x — mx — ny) x+ v m + (1 +»*- + «•) v-^ — (3). 



- /« (&=-«;') (r- j;^ = (4), 



(s — WA'— ?/y)y+ «-« +(1 + ?«- + //■) L- -^— = (5), 



f ^ 1 (1 + II) («■' - «^) + ( 1+ w.=) (// _ C-) + (>ir + «;■') (r - I'O } 



-u{a'-v'){c'-v') = (6). 



Uetweeii these six equations the five quantities m, n, v, -, — , -j-, are to 



be eliminated, and the resulting equation will be that of the wave 

 surface. These are the equations given by Fresnel, but lie was not 

 successful in effecting the requisite eliminations. 



The fundamental equations may be put under a symmetrical form 

 by the introduction of an additional symbol. 



If for m and n we substitute respectively — -, and , and suppose 



/. M. II. connected by the equation 



l^ + ni- + ii' = \, 

 we shall, instead of (3) and (4), have the three equations 



Ix + tni/ + }iz = V (1). 



/- + m- + H- = 1 (2), 



— — ? + — — n + —2 — 2 = (3). 



V — a V — b V - c ' 



Differentiate these equations with regard to /, m, n, and v. 



xdl+ ydm + zdn = dv (4), 



/ff/ + mdm + iidti — (.5), 



- — .dl+ ,'\.^ .dm+ /%rf« = |( .,— j) + i-^] + (-T^) l«f/('... (6). 

 v-fi V -h i)-c \\v--al Vc-ft"/ \r-c-l I ' 



