On Systems of Rays. 47 
and to the relations which result from these, by differentiation and elimination. For 
thus we obtain 
a yoQ SOQ .&w SO wv SQ .w 
ra” a 7 dc? © be * bss a. 
oor ire y (F’) 
5 te 12 ea 3 we A@) oe 0) 5 ov 
v dadu te * 3:55 “sp 2 by’ 
aie ey _ vy 62_ FO po fee 3 2 ole tty 
v ay. 7 ax ~dadx ca Ordx of dvdx dy 
in which v is considered as a homogeneous function of the first dimension of a, f, y, 
involving also the colour y ; and in which, although the three variations éa, 63, ey, 
are connected by the relation a&a+(03+ysy=0, yet we may treat these variations 
as independent; because, if we introduced indeterminate multipliers of ada + Pe + yey, 
in (Ff), to allow for the relation, we should find that these multipliers vanish, on 
account of the conditions of homogeneity of v. And if we put for abridgment 
“oe ae oQ >? FO FQ oO eee a 
— So? door rr Ss ot sa) +5 2 (2), (G ) 
wv 
the equations (F"°) give the following formula for &v, that is, for the second variation 
of v, taken as if a B y x were four weet variables, 
oy — — 08 =) a 
vw 
a+r tu 
a er) = 
a 208 = 9 (aa—o 2m 
a 
, 82 F8 = ao Ge vy =) 38 —v8" wet 
(Sv + 08"Q + 28v8Q) = 
ov: 
a = (38—o8 &) 20 (2)—vy 2) fea vy?) — (m—09 5) t 
“uate (a) =) au — 09") ESP (a- ee -w)t 
-arn $8 2 (aa—v 9 Se) 35 (28-08 se) (a) 29a) 
which justifies the passage from (D") to (H°), and expresses the law of dependence 
of the partial differential coefficients of the second order of the function v on those 
of Q, for the case of a uniform medium. 
If the medium be not uniform, and if we would still express the law of this depend- 
ence, we have onlp to change &, in the four equations (F"°) to a new characteristic 8,, 
