4 Professor Hamuton’s Third Supplement 
and if between the three first of these equations () we eliminate two of the three initial 
co-ordinates wv’ 7/' 2’, it is easy to perceive, by (C) or (D), that in every optical com- 
bination the third co-ordinate will disappear ; and similarly that between the three 
last equations (EZ) we can eliminate all the three final co-ordinates, by eliminating 
any two of them ; and that these eliminations will conduct to the relations (C) under 
the form 
O=Q (G, Ts Vy Ly Ys 2s X)s 
0= a) (a, T,U,2'5 y; z, Xs t (F) 
which can thus be obtained, by differentiation and elimination, from the characteristic 
function V” alone: and which, as we are about to see, determine the forms of », v’, 
that is, the properties of the extreme media. Comparing the differentials of the rela- 
tions (F’), with the following, that is, with the conditions of homogeneity of v, v', pre- 
pared by the definitions s ) a a ch relations (B), 
Pa Bt ay = as + Br + yp, ' é) 
Y= “+B a+ py ee cde 
and with their nen that is with 
ov sv, ov bv 
aba + Bor ae ews) +582 +3 OX 
/ i} H 
ado’ + for’ oF you’ = = Sh ad aE ae 
we find 
a_o2 P_&Q y_ ea 
yoo? 0 oF’ v dv? 
a’ _ 6a’ p __ 60! 7 _ 6a’ t (1) 
vy a! ? os Se wv SSN 
and also 
—le&_e2 -1&_e&@ —lov_82 —1_ oa 
ode Se? voy dy? vdz dz? vdy dy’ 
=13¥ Bo! 1 Be 8a 1 ae a" 1 be _ 80 
Dir ea Od es ennai: iG ay? v df 8”? 3x aon 
ac a 0 aman 4 (L) 
a ets |) 
which can be done by putting those relations under the form 
0 = (co? +7? 2) 2 — 
Ca) 1 en (M) 
O= (6? +7? +2)? w —-1=0'; 
