On Systems of Rays. 11 
in which A is the common yalue of the three first equated quantities in (J/’), and X’ is 
the common value of the three equated quantities in(O’). And in this case, by 
means of the two equations (O'), and the two that remain of (JZ), combined with 
the two relations (J<') (NV’), we can eliminate o, 7, v, o,7,v, from (F"), and so deduce 
PV from T: while, in the same case, or even if the initial medium alone be uniform, we 
are to deduce /V from T, by eliminating o’, x’, v', between the equations (7’)(N’)( 0’). 
When all the media of the combination are not only uniform, but bounded by 
plane surfaces, which happens in investigations respecting prisms, ordinary or extra- 
ordinary, then of the seven quantities o, r, v, «,7',v, x, only three are independent ; 
two other relations existing besides (KX) and (V’), which may be thus denoted, 
0=o7 (o, Ty Uy Oy T5 Vy x)» ; 
0=07 (o, 7, v, OyT,0U5 x) > t (Q) 
because, in this case, the initial direction, and the colour, determine the final direc- 
tion. In this case, we may still treat the variations of «, 7, v, o', 7, v, x, as indepen- 
dent, in 87, by introducing the variations of the four conditions (A’) (N’) (Q), 
multiplied by factors A, d’, \”, \’, that is by putting 
8 T= 280 —2'8o' +ydr — yor + 280 — 2/80 — > x 
+ ASQ + N'8Q' +2"8Q" +2"80": CR’) 
an equation which decomposes itself into the seven following, 
sT so, 80 on 82" 
eel ad cee iit a ee 
8T Sanewsa” 30” 
SN aAY RA iN” 
8T 8a » 0D Oe 
ai 2 = AS +X Sig +2 Su : 
oT fi len OQ “ 6Q” m 6a” / 
3° 12 = Par ix Ww +2 Be? (S) 
SPM OW RS Sarva MBry wea” 
ge YSN G IN GE AG 
Sth ce eel Syn OO, SNe 
5 7 z= r 3 7 +X SV + r Te > 
T V ¢ “u" vu 
7 oh ees \ re) x oQ 17 02" 7 2" 
By eT TT By TO? 
between the six first of: which, and the five equations marked (F’) (K’) (N’) (Q’), 
we can eliminate the ten quantities o, 7, v, 0,7, v,A, A’, X’, ”, and thus deduce the 
relation between V, x, y, z, x’, y', 7, x, from that between 7; o, 7, v, 6, 7, v’, x. It 
is easy to extend this method to other cases, in which there exists a mutual depend- 
ence, expressed by any number of equations, betwen the seven quantities o, 7, v, 
Ty T,V5 Xe 
And all the foregoing details respecting the mutual deductions of the functions 
