On Systems of Rays. 67 
of a given ordinary or extraordinary path, and that the positive semiaxes of 2, 2’, 
coincide with the final and initial directions, so as to give 
B= 0/0570) a=0N oO. — 5 ov 0), 
/ - , i U i , : i (B’) 
2 =0, 7 =0,2=0, d=0,8.=0, y =1, sy =0; 
and then the six equations (.4°), of which only four are distinct, reduce themselves 
to the four ae 
OV 
o2v i OV F OV ev 
Sano Ve 3B pian seem Oo (ae ee x } 
eV 8 SV Sy do | 
¥ i sel po (erage ape (=- Sais) bea 
Sv sue {ay 4 OV ’ OV ow 5 
sap +3 P= gar & + Har Y + (Ram apH) & | 
eV ov eV &v cv dt 
o eemarey ar+ (Se sma) y+ (5 — saz) 23 2 er mes, 
a Sv! Lee sy OV OV coy 
oar 8a'sp 2. Sede! + 5y8. ' oy + (a7 Sy oa'd x | 
SF Sv! eV Soy, Se! Be 
a i a oe oa! + ( ba'dy! + ae ar ee da'dz! a) a 
Se! So! Se SV SF fy 
ay ft 1 BY 
Sap" ~ spr °F = pay®* t Hay Yt Leste see) x a 
OV ov! } SF Su! o oe 
=F (soak ae ou! + (52 + Spay’ a} sy — = —sg52) © 
they give therefore, by easy eliminations, expressions for éa, 03, ea’, @f', of the form 
6 O o Sov & owe Qaee edte 
ea= 5, ety + et ee + Hi BN } 
38 = 2 ar + ze Cys oz + 2 aa! + ey wc 
Z (D’) 
ee ere ar + y + = By, | 
apa a + Fy + Bee En E ees: J 
which involye twenty-four coefficients, and enable us to determine the general 
geometrical relations between the final and initial tangents to the near luminous paths. 
If the extreme media be ordinary, that is, if the functions v, v', be independent of 
the directions of the rays, we have 
V=pV/ (a” + ice + y)s v = im Jf (a’* + p° ata ys (E’) 
