78 Professor Hamitton’s Third Supplement 
will have four sections equal and similar to this final locus, namely, the sections by the 
four planes (F'""). We may therefore consider these as four guiding planes for the 
initial ray, since each contains for any proposed final curve or locus (G") of the final 
point B;, an equal and similar guiding curve or locus, which is a section of the 
sought initial cone, and by which therefore that cone may be determined. If, then, 
we know these four guiding planes, or any one of them, and the corresponding sys- 
tem of final and initial rectangular directions, or conjugate guiding axes, of which 
two are determined by a guiding plane, we shall be able to construct the imitial ray- 
line or ray-cone corresponding to any final position or locus of the point B,. The 
fourth and last general problem of those proposed above, may therefore be considered 
as resolved, by this theory of the guiding planes and guiding axes. 
We see then that in order to compare completely the extreme directions of any 
two near luminous paths 
(A, B),.s (A, BAYS, &3 
in which 4 is the initial and B the final point of a given path, and 4’, B’, are any 
other initial and final points infinitely near to these, the following geometrical elements 
of arrangement, or some data equivalent to them, are necessary and sufficient to be 
known. 
First. The final axis, and the initial axis, of chromatic dispersion ; and the corres- 
ponding final and initial constants & &, with their proper signs, to indicate the direc- 
tions, as well as the quantities of dispersion. 
Second. The final axis, and the initial axis, of curvature of the given path. 
Third. The final pair, and the initial pair, of axes of curvature of the guiding 
paraboloids, at the ends of this given path; and the final and initial constants of 
deyiation 7, 7’. 
Fourth. A guiding plane for the initial ray-lines, and a guiding plane for the final 
ray-lines ; together with the final system and the initial system of rectangular direc- 
tions, or conjugate guiding axes, connected with these guiding planes. 
When these different elements of arrangement of the extreme ray-lines are known, 
we can deduce from them the dependence of &a, 88, da’, 88’, and more generally of 
da, 83, dy, ba’, 88, dy’, on aa, dy, Sz, dx’, dy’, dz’, dy ; and reciprocally when this latter 
dependence has been deduced from the partial differential coefficients of the charac- 
teristic or related functions, we can deduce from it the geometrical elements above 
mentioned. 
