On Systems of Rays. 3 
properties of the final and initial media ; but these final and initial mediwm-functions 
v, vw, may themselves be deduced from the one characteristic function ’, by reasonings 
of the following kind. 
Whatever be the nature of the final medium, that is, whatever be the law of 
dependence of v on the position, direction, and colour, we have supposed, in deducing 
the general formula (4), that the expression of this dependence has been so prepared 
as to make the medium-function v homogeneous of the first dimension relatively to 
the direction-cosines a, 3, y ; the partial differential co-efficients 
ov oe Ou 
8a’ 3B” dy’ 
of this homogeneous function, are therefore themselves homogeneous, but of the 
dimension zero ; that is, they are functions of the two ratios 
a 
ry? 
involving also, in general, the co-ordinates « y z, and the chromatic index y: if then 
we conceive the two ratios 
to be eliminated between the three first of the equations (B), and if, in like manner, 
we conceive 
to be eliminated between the three last equations (B), we see that such eliminations 
would give two partial differential equations of the first order, between the character- 
istic function V’ and the co-ordinates and colour, of the form 
V&V BV 
= « ue Se De wi 
O= Q’ eV ov OV , , , (C) 
= (spp gpe ge oes x )» 
which both conduct to the following general equation, of the second order and third 
degree, common to all optical combinations, 
BY OE og ui ip an FV apie Ye ilo AK hase 
oxen §dydy' = 8z82’ i ardy’ dydz Bzer’ t ra? 3 yea’ dézéy’ 
< _ Oe CIR Oe eV 8V SV SV SF &V 
—Ssde’ Syd Sade! | Med ByeT Bede ted Gye’ Busy" 
If now we put, for abridgment, 
Dil SIE UBT 
ww —<9; ay —T; % =U, 
ras Lei ae Le 8 Rie 
(D) 
