82 Professor Hamitton’s Third Supplement 
real) are tangents to the two caustic surfaces ; that is, to the two surfaces which are 
touched by the final rays, and are the loci of the two points of vergency. If the 
guiding lines are imaginary then the points of vergency are so too, and the final rays 
are not all tangents to any common surface. We shall have occasion to resume here- 
after the theory of the caustic and developable surfaces. 
If it happen that 
; t—r=+2n, (v") 
without ¢—r and separately vanishing, then the two planes of vergency close up 
into one plane, bisecting one pair of the right angles formed by the two principal 
planes of curvature of the guiding paraboloid ; the two vergencies reduce themselves 
to a single vergency, corresponding to this single plane, and equal to the semisum of 
the two curvatures of the same surface: and the two guiding lines reduce themselves 
to a single guiding line, passing through the corresponding point of convergence or 
divergence, and having still the property of being intersected by all the near final 
ray-lines, although this property is not now sufficient to determine this system of ray- 
lines. 
But if the two members of (/”") vanish separately, that is, if the difference of 
curvatures and the constant of deviation are separately equal to zero, then the guiding 
paraboloid is a surface of revolution, having its summit at the given final point B, and 
all the near final ray-lines are normals to this paraboloid of revolution, and (with the 
same order of approximation) to the osculating sphere at its summit, and they all pass 
through the centre of this sphere. Reciprocally, if there be any one point 0, 0, Z, 
through which all the final ray-lines pass, the equations (JC) give 
=O t=T= 4 CW) 
and the more general equations (7°), in which the rectangular axes of x and y are 
arbitrary, give 
da _ $B _ —1. 8a _ = a 5 vil 
may ee 5 Aig OF vagreee ee) 
that is, by (G°), or (C*), 
Py ee eee ee 
3a? Sa? Sadr.” 
ov Z- Og Or epee z 
Sriy |” dadB Sady  SGdz° @) 
eV Se ADtly ik Desa 
ay + 2 BB — Spay’ 
When the final rays are straight, and satisfy these last conditions (¥"), which then 
reduce themselves to the following, 
