58 Professor Hamitron’s Third Supplement 
Three of these seven equations into which (U") resolves itself, give, by a proper 
combination, a value for the trinomial 
s Sox Nae Sr. Sy Ou bz’ 
which enables us to eliminate that trinomial from (U") and so to deduce a value for 
aA, which being combined with (FR) gives, 
2 Ao 80, SQ. / &V, Su 
Gs } Ge New a ae = on i daa) 
= 80, 80, / 87, S 
atcok es rar) * 25, eee) 
2 \2 (8M j dQ, 80, /& & ky 
(= ) (= - ee) 24 Oe: = (Set =) 
80, 80, 20, 80; | 80, 30) 
as a a Sa FEE 
SQ. du SQ, du Qs, du 
io) ot 
(eesone a Sy | Bus 5) 
6Q, oy ot\  dQe oF; ou\ Qe 8"; ou 
Sz ( . oe ine =e as ( 8 oy +A8 a) ( ) 
OQ 6Qs . Qs = 
+ 
bax 
Laas yrs oz + 
=(8V,—8V —d&'u) 
3X 
(2. &u ne ou 8D. li (v") 
Ora dy ‘ 6u2 oz 
a formula that is equivalent to twenty-eight separate expressions for the twenty-eight 
coefficients of 772, of the second order. This formula supposes the rays to be reflected 
or refracted into a variable medium; but it can be adapted to the simpler supposition 
of reflexion or refraction into an uniform medium, by merely making the quantities 
éQ, 60, rere 
ox ‘ oy : en 
formula (V") gives, 
; vanish, Whether the last medium be variable or uniform, the 
8°, =8°V, ; (W’) 
8 referring, as in former numbers of this Supplement, to the variations of 2’, 7/, 2’, x, 
alone, that is, to the variations of the initial co-ordinates and of the colour ; and the 
final co-ordinates x y 2 being those of any point on the reflecting or refracting sur- 
face. Thus the ten differential coefficients, of the second order, of the characteristic 
function V, like the four of the first erder, taken with respect to the initial co-ordi- 
nates and the colour, undergo no sudden change by reflexion or refraction ; but the 
differential coefficients of both orders, which involve the final co-ordinates, take sud- 
denly new values which we have shown how to determine : and from these new coefti- 
