4A Professor Hamiiton’s Third Supplement 
of uniform media. And if the extreme media be not uniform, but variable, that is, 
if they be atmospheres, ordinary or extraordinary, we can still connect the partial 
differential coefficients of the three functions, by the general method mentioned at 
the beginning of the seventh number: which method extends to orders higher than 
the second, without much additional difficulty of elimination, but with results of 
greater complexity, and of less interesting application. 
This general method consists, as has been said, in differentiating and comparing the 
equations into which the general expressions (4') (B’) (C’) for the variations of the 
three functions resolve themselves : and in making this preliminary resolution of the 
general expressions (A’) (B’) (C’), it is necessary to attend with care to the rela- 
tions between the variables oc, 7, v, 6, 7, v, x, or between o, 7, v, v, 7’, 2, x, when 
any such relations exist. The investigations into which we have entered im the three 
last numbers, for the case of extreme uniform media, suppose that the variables are 
connected only by the relations Q=0, Q'=0, which arise from and express the optical 
properties of these media; and other but analogous processes must be deduced from 
the general method, when any additional relations Q" =0, Q" =0,... between the 
variables of the question, arise from the particular nature of a combination which we 
wish to study. In the very simple case, for instance, of a single uniform medium, 
we have the three relations 
o=0, T=T, v=, (U’) 
which are to be combined with the relation Q=0; and with this combination of rela- 
tions, the general expression (C’) for the variation of 7’ will no longer admit of being 
resolved in the same way as when more of the quantities on which 7” depends could 
vary independently of each other. 
In the case last mentioned, of a single uniform medium, the characteristic function 
V’ involves the co-ordinates x, y, 2, 2’, y/, #, only by involving their differences 7 —2", 
y—y, 2-2, and is, with respect to these differences, homogeneous of the first 
dimension, being determined by an equation of the form 
Oh = i > fee ? s, X ); G¥®) 
which results from the equation (IV) for the medium function v, by first suppressing 
in that equation the co-ordinates on account of the supposed uniformity, and then 
making 
a. 0-2. Bs Ws op. 2S2! 5 
= (W") 
The relation (V°) may also be deduced from the relation Q=0, by eliminating the 
ratios of o, r, v, between the three following equations, 
