On Systems of Rays. 97 
the corresponding plane of deflexure. And when the compensation and therefore 
the intersection becomes possible, by the constant of deviation being less than the 
semidifference of the two extreme deflexures, then the two real planes of vergency of 
the near final ray-lines from the oblique plane are symmetrically situated with respect 
to the two rectangular planes of extreme deflexure : which latter planes may also, for 
a reason already alluded to, be called the planes of extreme projection of the final 
ray-lines. 
Other Geometrical Illustrations of the Condition of Intersection, and of the Elements 
Arrangement. Composition of Partial Deviations. Rotation round the Axis of 
Curvature of a Final Ray. 
20. The condition of intersection of two near final ray-lines may also be illustrated, 
and might have obtained, by other geometrical considerations, on which we shall 
dwell a little, because they will help to illustrate and improve the theory of the 
elements of arrangement. 
It was remarked, in the fourteenth number, that the general comparison of a given 
luminous path (4, B), with a near path (4’', B’), +, might be decomposed into 
several particular comparisons, such as the comparisons with the less general near paths 
(A, B,),, (A, B;),, and others, on account of the linear form of the expressions (D") 
for the variations Sa, 83, 8a’, 88’, of the extreme small cosines of direction, which form 
permits us to consider separately and successively the influence of the variations of 
the extreme co-ordinates and colour, or the influence of any groupes of these varia- 
tions. Accordingly, by an Analysis founded on this remark, we decomposed the 
general discussion of the geometrical relations of infinitely near rays into four less 
general problems, which were treated of, in the fifteenth number. The applications, 
in the sixteenth number, to questions respecting the mutual intersections of the final 
ray-lines from the final perpendicular plane, may be considered as only illustrations 
and corollaries of the third of those four problems: but the questions since discussed, 
respecting the ray-lines from an oblique plane, require a combination of the solutions 
of the second and third of the four problems, and furnish therefore, an example of 
the Synthesis of those elements of arrangement of near rays, to which the former 
Analysis had conducted. This synthesis, however, has in the foregoing numbers been 
itself algebraically performed, (namely, by the algebraical addition of certain partial 
variations, ) although many of the results were enunciated geometrically, and com- 
bined with geometrical conceptions: but a geometrical idea and method, of the Syn- 
thesis of the Elements of Arrangement, may be obtained by considering, in a general 
manner, the geometrical composition of partial deviations. 
VOL. XVII. 2G 
