120 Professor Hamitton’s Third Supplement 
of luminous paths, polygon or curved, of any common colour, which undergo any num- 
ber of refractions or reflexions, ordinary or extraordinary, and gradual or sudden, it 
is easy to see that we may consider these paths as touching a certain set of caustic 
curves, in the final state of the system, and therefore as grouped into certain sets of 
consecutively intersecting paths, and as having for their loci certain corresponding 
sets of ray-surfaces, which may be called caustic pencils: and that these caustic 
pencils are touched by the transition planes (O"), while the transition-points 
(P") are on the caustic curves, and therefore on their loci the caustic surfaces. 
The transition-points are also evidently the points of consecutive intersection, or of 
vergency, of the luminous paths from 4, in the final state of the system. And it is 
manifest, from the foregoing remarks, that these final points of intersection are also 
transition-points in the following other sense, that when the point C, in moving along 
the prolongation of the path (4, 6), arrives at any one of these positions of inter- 
section, the condition of existence of maximum or minimum of the function V+ V, 
begins or ceases to be satisfied. 
The separating planes P’ P”, have, when real, another remarkable property, 
namely, that of containing the directions of mutual osculation, at the point B, of the 
two action-surfaces or waves determined by the equations 
V=const., V,=const.; (R”) 
for these equations may be put approximately under the following forms, (when we 
choose the point B for origin and the final direction of the path (4, B) for the 
positive semiaxis of z, so as to have a=0, B=0, y=1,) 
z=petqy +4hra’+sxy +h ty’, (s") 
Z=pe+qy thre tsay they, 
in which the coefficients have the following relations, 
P,=P, 7% 
_ 1/8¥, sv 
ne ee Gases 
an & Ve eV (GES) 
Si aay (say ae, : 
1 SV, 37 
4—t= = Gr +50) 
and therefore the planes 
O=(r, —1) 2° +2(s,-s)ayt+(4—-by’, (U"’) 
which pass through the given ray-line at the point B, and contain the directions of 
osculation of the second order of the two touching surfaces (2) or (S"), are the 
