On Systems of Rays. 79 
Application of the Elements of Arrangement. Connexion of the two final Ver- 
gencies, and Planes of Vergency, and Guiding Lines, with the two principal 
Curvatures and Planes of Curvature of the Guiding Paraboloid, and with the 
Constant of Deviation. The Planes of Curvature are the Planes of Extreme 
Projection of the final Ray-Lines. 
16. To give now an example of the application of these geometrical elements of 
arrangement, let us employ them to determine the conditions of intersection of two 
near final ray-lines, corresponding to a given colour and to a given luminous origin ; 
and let us suppose, for simplicity, that one of these two straight ray-lines being the 
final portion or final tangent of a given luminous path (4d, B),, the other corres- 
ponds (as in the third of the foregoing problems) to a final point B, on the given 
final plane perpendicular to this given path at B. Then if the constant 7 of devia- 
tion vanishes, so that the final ray-lines are normals to the guiding paraboloid, the 
condition of intersection requires evidently that the near point B, should be in one 
of the two principal diametral planes, that is, on one of the two rectangular tangents 
to the lines of curvature on this surface ; and the corresponding point of intersection 
must be one of the two centres of curvature. But when m does not vanish, the 
deviation of the ray-lines obliges us to alter this result. The intersection of the near 
ray-line with the given ray-line will not now take place for the directions of the lines 
of curvature ; but for those other directions, if any, for which the angular deviation 
nel of the ray-line from the normal is equal and contrary to the angular deviation of 
the normal from the corresponding plane of normal section, that is, from the corres- 
ponding diametral plane of the guiding paraboloid. This latter deviation, abstract- 
ing from sign, is, by the general properties of normals, equal to the semidifference of 
curvatures multiplied by the element of the normal section &/, and by the sign of 
twice the inclination of this element to either of the lines of curvature; it cannot 
therefore destroy the deviation of the ray-line from the normal, unless the semi- 
difference of the two principal curvatures of the paraboloid is greater, or at least not 
less, abstracting from sign, than the constant of deviation 7; this then is a necessary 
condition for the possibility of the intersection sought. But when the semidifference 
of curvatures is greater (abstracting from sign) than n, then there are two distinct 
directions P,, P,, of the normal or diametral plane of section, symmetrically placed 
with respect to the two principal planes of curvature, and such that if the element of 
section o/ be contained in either of these two planes, P,, P,, (but not if the element 
8 be in any other normal plane,) the corresponding ray-line from the extremity of 
that element will be contained in the same normal plane P, or P,, and will intersect 
the given ray-line as required ; and the point of intersection of these two near ray- 
