On Systems of Rays. 53 
fu du bu 
dz’ Sy? 82’ 
of incidence, and the normal to the reflecting or refracting surface at that point. A 
remarkable case of indeterminateness, however, or rather two such cases, will appear, 
when we come to treat, in a future number, of external and internal conical refraction. 
With respect to the new form V’, of the characteristic function V’, it is to be deter- 
mined by the two following conditions ; first, by the condition of satisfying, at the sur- 
face (B"), the equation in finite differences (4’), that is, by the condition of becoming 
equal to the value of the old form V,, when the final co-ordinates x, y, z, are con- 
nected by the relation w=0; and secondly by the condition of satisfying, when the 
if we knowalso y, 2, y, Z, and the ratios of that is the colour, the point 
final co-ordinates are considered as arbitrary, the partial differential equation of the 
form (C), 
en SaMe ae ale «ae 7 
020, (EB swnx), 
if the final medium be variable, or the simpler partial differential equation of the 
form (/”’), if that final medium be uniform. And as it has been already shown that 
the partial differential equations relative to the characteristic function V, may be 
transformed, and in the case of uniform media integrated, by the help of the auxiliary 
functions JV, J, it is useful to consider here the changes of those auxiliary func- 
tions, which are also otherwise interesting. 
It easily follows from the definitions of W, JT, that the increments of these two 
functions, acquired in reflexion or refraction, are equal to each other, and may be 
thus expressed, 
AW=AT=zZAc +yAr +ZzAv. (G’) 
And because the differences Ac, Ar, Av, are, by the general equations of reflexion or 
ou du du 
refraction (D"), proportional to Be? By? 32” we may consider these differences as 
equal to the projections, on the rectangular axes of the co-ordinates x, y, z, of a 
straight line = v(Ao*+Ar*+Av*), perpendicular to the reflecting or refracting sur- 
face at the point of incidence, and making with the axes of co-ordinates angles of 
which the cosines may be called 7,, ,, 2, ; in such a manner that we shall have 
Ao=n, J (Ac* + Ar* + Av?) ; 
Ar=n, J (Ao? + Ar + Av’); 
Av=n, / (Ao? + Ar? + Av’); 
AW=AT=(an, +yn, + 2N,) / (Ao* + Ar? + Av’). 
(H’) 
Now the quantity an, +yn,+zn, is equal, abstracting from sign, to the perpendicular 
let fall from the origin of co-ordinates on the plane which touches the reflecting or 
refracting surface at the point of incidence ; it is therefore constant if that surface be 
VOL. XVII. P 
