122 Professor Haminton’s Third Supplement 
the points B, C, are both in one common uniform medium, so that the paths 
(B, C) (B, C) are straight, then each of the caustic pencils, or ray-surfaces, 
composed of such straight paths consecutively intersecting each other and touching 
one caustic curve, becomes a developable pencil, and its tangent plane becomes a 
plane of vergency, of the kind considered in the sixteenth number. The relations 
also between the two planes of vergency in a final uniform medium, which were 
pointed out in the twenty-first number, may easily be deduced from the present more 
general view and from the recent theorems of osculation ; for thus we are led to con- 
sider a series of waves or action-surfaces /”,, similar and similarly placed, and deter- 
mined in shape but not in size or focus by the uniform medium, and then to seek the 
extreme or limiting surfaces of this set which osculate to the given surface ” at the 
given point B; and since it can be shown that im general among any series of sur- 
faces, similar and similarly placed, but having arbitrary magnitudes, and osculating 
to a given surface at a given point, there are two extreme osculating surfaces, real 
or imaginary, and that the tangents which mark the two corresponding directions of 
osculation are conjugate tangents (of the kind discovered by M. Dupry) on each sur- 
Sace of the osculating series, and also on the given surface, it follows as before that 
the conjugate planes of vergency in a final uniform medium are conjugate planes of 
deflexure of each medium-surface /”, and also of the surface /” determined by the 
whole combination. When the final medium is ordinary as well as uniform, then the 
osculating surfaces /”, are spheres, and the directions of extreme osculation are the 
rectangular directions of the lines of curvature on the surface /, which is now per- 
pendicular to the rays ; in this case, therefore, and more generally when a given final 
ray in a final uniform medium corresponds to an umbilical point or point of spheric 
curvature on the medium-surface /”,, the planes of vergency cut that surface, and the 
surface Y” to which it osculates, in two rectangular directions, because two conjugate 
tangents at an umbilical point are always perpendicular to each other: and, in like 
manner, the planes of vergency being conjugate planes of deflexure will (by the 
seventeenth number) be themselves rectangular, if the final ray whether ordinary or 
extraordimary be such that taking it for the axis of deflexion of the medium-surface 
V the indicating cylinder of deflexion is circular. 
The foregoing principles give also the law of osculation of the variable medium- 
surface /”, between its extreme positions, in a final uniform medium, namely, that the 
distances of the variable osculating focus from the two points of vergency, are pre- 
portional to the squares of the sines of the inclinations of the variable plane of oscu- 
lation to the two planes of vergency, multiplied respectively by certain constant 
Jactors. A formula expressing this law was deduced in the First Supplement ; but 
the constant and in general unequal factors, (in the formula Z and 1,) for the squares 
of the sines of the inclinations, were inadvertently omitted in the enunciation, Our 
present methods would enable us to investigate without difficulty the law for the more 
