16. 
17. 
20. 
21. 
- Calculation of the elements of arrangement, for arbitrary axes of co- soraihetee” c 5 
25. The general linear expressions for the arrangement of near rays, fail at a point of vergency. ane. 
vill 
P 
. Discussion of the four problems. Elements of arrangement of near luminous paths Axis and 
constant of chromatic dispersion. Axis of curvature of ray. Guiding paraboloid, and constant 
of deviation. Guiding planes, and conjugate guiding axes..........s.ceeeeeeeeeeeeeeeneee soneeeees 
Application of the elements of arrangement. Connexion of the igs final pettericted eid ae of 
vergency, and guiding lines, with the two curvatures and planes of curvature of the guiding surface, 
and with the constant of deviation. The planes of curvature are the planes of extreme projection 
Ofsthe fittal Fay-UNESs.iosnccucs wweescers veces tdsbonee svoncsnasctentoncessenclsts covedtueatebamencienee cess chest ee: 
Second application of the elements. Arrangement of the near final ray-lines from an oblique plane. 
Generalisation of the theory of the guiding paraboloid and constant of deviation. General theory 
of deflexures of surfaces. Circles and axes of deflexure. Rectangular planes and axes of ex- 
treme deflexure. Deflected .ines, passing through these axes, and having the centres of deflexure 
for their corresponding foci by projection. Conjugate planes of deflexure, and indicating cylinder 
OP GEMEKION <2. s.cnseceevenissncoclovsacevscsctanescesaceeesetectenv ener seapedesacdsusetese sav aacaesis sacs ee neeNer 
. Construction of the new auxiliary paraboloid, (or of an osculating hyperboloid,) and of the new 
constant of deviation, for ray-lines from an oblique plawe, by the former elements of arrangement. 
- Condition of intersection of two near final ray-lines. Conical locus of the near final points in a 
variable medium which satisfy this condition. Investigations of Matus. Illustration of the con- 
dition of intersection, by the theory of the auxiliary paraboloid, for ray-lines from an tacaatd 
plane: 83, .00.29% He, FES aoe 
Other geamictiieal aiinatraGons of thes conditied ‘of ifitersedtions niet of ie clined of arrangement. 
Composition of partial deviations. Rotation round the axis of curvature of a final ray............ 
Relations between the elements of arrangement, depending only on the extreme points, directions, 
and colour of a given luminous path, and on the extreme media. In a final uniform medium, 
ordinary or extraordinary, the two planes of vergency are conjugate planes of deflexure of any 
surface of a certain class determined by the final medium; and also of a certain analogous surface 
determined by the whole combination. Relations between the visible magnitudes and distor- 
tions of any two small objects viewed from each other through any optical combination. Inter- 
changeable eye-axes and object-axes of distortion. Planes of no distortion.. 
termination of these points, and of their loci the caustic surfaces, in a straight or curved system, 
by the methods of the present Supplement .. 
- Connexion of the conditions of initial and final initeodec tier of 4 “ey near acenie of Aree’ Berea or 
curved, with the maxima or minima of the time or action-function V+ V,= 3 fvds. Separating 
planes, transition planes, and transition points, suggested by these maxima and minima. The 
separating planes divide the near points of less from those of greater action, and they contain the 
directions of osculation or intersection of the surfaces for which V and V, are constant; the 
transition planes touch the caustic pencils, and the transition points are on the caustic curves. 
Extreme osculating waves, or action surfaces. Law of osculation. Analogous theorems for 
sudden reflection or refraction .......... sdeuccenesceevscerecneesuseeses 
25. Principal rays and principal foci for strat lie or ited syeieuts General craked? of aetaathing 
the arrangement and aberrations of the rays, near a principal focus, or other point of vergency... 
26. Combination of the foregoing view of optics with the undulatory theory of light. The quantities 
o, T, UV, or 
ov 8V SV 
Sz? by 2 Se? 
that is, the partial differential co-efficients of the first order of the characteristic function V, 
taken with respect to the final co-ordinates, are, in the undulatory theory of light, the compo- 
nents of normal slowness of propagation of a wave. The fundamental formula (A) may easily 
be explained and proved by the principles of the same theory.......ccscescecceseeseeereereeenseneenens 
age 
71 
79 
83 
90 
97 
101 
..108 
Roll tit 
116 
123 
