On Systems of Rays. 123 
complicated case, when the osculating focus C’ being still in a uniform medium, the 
point of osculation B is in another uniform medium, or even in an atmosphere ordi- 
nary or extraordinary. 
We might extend the reasonings of the present number to the case of sudden 
reflexion or refraction, ordinary or extraordinary, and obtain analogous results, which 
would include, in like manner, the results of former memoirs. In this case we should 
find a certain analogous condition for the existence of a maximum or minimum of the 
function = fvds ; and when this condition is not satisfied, we should have to consider 
two pairs of separating planes, which cross the tangent plane of the reflecting or 
refracting surface in one common pair of separating lines: the two pairs of planes 
passing together from the real to the imaginary state, and in this passage closing up 
into two transition-planes, which touch the caustic pencil before and after the sudden 
reflexion or refraction, and intersect in one common ftransition-line, on the tangent 
plane of the reflector or refractor, connected with a transition-point upon the caustic 
curve of the pencil, and with certain extreme osculating waves or action-surfaces and 
focal reflectors or refractors, of a kind easily discovered from the analogy of the 
foregoing results. 
Formule for the Principal Foci and Principal Rays of a Straight or Curved System, 
Ordinary or Extraordinary. General method of investigating the Arrangement 
and Aberrations of the Rays, near a Principal Focus, or other point of vergency. 
25. Among the various points of consecutive intersection of the rays of an optical 
system, there are in general certain eminent points of vergency, in which certain 
particular luminous paths are intersected each by all the infinitely near paths of the 
system. These eminent points and paths have been pointed out in my former 
memoirs, and have been called principal foci, and principal rays. They may be 
determined for straight final systems, by the characteristic function V7, and by any 
three of the six following equations, 
&Vv 1 dv OV 1 Sy 
a2 t+ Roe — Saby | R Sadp— ; 
eV | sy ov 1 82v 
aye Rpt Bydz TR Hay ihe 
OFA EAU OLS ov 1 38% 
a2 TRO 7° Bact Roe 
x, y, 2 being the co-ordinates of any point on a principal ray, and a +ah, y+PR, 
z+yR being the co-ordinates of the principal focus; they may a!so be deduced from 
