52 Professor Hamitton’s Third Supplement 
tion given in the First Supplement, of the sudden changes produced in these func- 
tions and in their coefficients, by reflexion or refraction, ordinary or extraordinary. 
The general formula of such changes, which easily results from the nature of the 
characteristic function V, is 
0O=AV=V,—V,; (A’) 
V,, V., being the two successive forms of the function V, before and after the refiex- 
ion or refraction ; and the final co-ordinates x, y, z, im these forms, being connected 
by the equation 
O=u (a; Ys Z) (B’) 
of the reflecting or refracting surface. The formula 4’) may be differentiated any 
number of times with reference to the final and initial co-ordinates and the colour, 
attending to the relation (") ; and such differentiation, combined with the properties 
of the final uniform or variable media, conducts to the general laws of reflexion and 
refraction, and to all the conditions necessary for determining the changes of the 
coefficients of 7, and therefore also of the connected coefficients of JV and T, as 
well as to the laws of change of the functions V, JV, 7, themselves. 
Thus, for the first order, we have the general formula 
8V.—8V,=sAV =dou, €@) 
which, on account of the multiplier A, and the definitions (#7), resolves itself into the 
seven following, 
ou ou ou 
Ao= AT 3 ATS 5 Av= r= 5 
. av Ge 
Ag =Oigh Ar =10.5) Av =05 Ay =0: 
xX 
the symbol A referring, as in (4’), to the finite changes produced at the surface (B’), 
so that Ac, Ar, Av, denote the differences 6,—0,, t2.—7), v.—v,, between the new and 
the old values of o, 7, v, that is of the partial differential coefficients of the first order, 
of the characteristic function V, taken with respect to the final co-ordinates. The 
three first of the equations (D") contain the general laws of the sudden reflexion or 
refraction of a straight or curved ray, ordinary or extraordinary ; because, when com- 
bined with the equation of the form (F’), 
O=Qr (on, To, Uo5 Ly Ys 2; x)» (E’) 
which expresses the nature of the final medium, they suffice, in general, when that 
final medium is known, to determine, or at least to restrict to a finite variety, the 
new values o,, 72, v2, of the quantities c, 7, v, on which the direction of the reflected 
or refracted ray depends, if we know the old values o, 7, v,, which depend on the 
direction of the incident ray and on the properties of the medium containing it, and 
