On Systems of Rays. 21 
But if the final medium be uniform, then o, z, v, x, are not independent, but related 
by (<'); and the formula (#) resolves itself, not into the seven equations (4G) and 
(H') but into the six equations (G) and (J), the differentials of which are to be 
combined with the differential of the relation (J<’), so as to give the expressions for 
80, Or, Ou, da, Ory dv, 8 x , which are to be compared with (4%) as before. And in 
this case, of a final uniform medium, we may employ, instead of the two first equa- 
tions (/’), any of the transformations of those equations in the foregoing number ; 
or we may employ the following transformations of (J’), 
L+z ov OW. +2 ov peo: Zz ov ee OW | B’ 
Sr ee re a i ee 
in which, WV is considered as a function of the six independent variables o, 7, y, 
x,y’, 2, obtained by substituting for v its value as a function of o, 7, x; the form of 
which function v depends on and characterises the properties of the final medium, 
and is deduced from the relation (AK). It may be useful here to go through the 
process last indicated, both to explain its nature more fully, and to have its results 
ready for future researches. 
Differentiating therefore the two first a (B*), we obtain 
bv WwW (ew ew 
bu +s oz — os + Zz corey os (-? =) 8a + (—- zo” )ar, Ae 
3 es 7 =(<" Bs) 4 ee gos () 
Be ein | sale taee sapere acasnW Se 187 2 
in which we have put for abridgment 
OW ew iy ow Ow , ow 
S eSeaeer ee? ote Saag vt’ Sag : 
5g URE Sie STs SSW, ys STE Am) 
See Lye aaa ae ee 
8 referring only to the variations of the initial co-ordinates and of the colour: and 
if we put 
ow 
i ‘(a = = ey i (egy (E°) 
the equations (C*) give, by elimination, 
w" so = v2 (ax +223 gor 255°) 
bs mao ter 25x): 
w" be = — 2) (ay + Be oo #55 8x) .* 
2 es Bee ee a +a Pedy ); 
VOL. XVII. G 
