96 Professor Hamitton’s Third Supplement 
especially because, in the former theory, the coefficients which mark the direction of a 
ray were left as independent and unconnected functions, whereas, in the latter, they 
are shown to be connected with each other, and to be deducible by uniform methods 
from one characteristic function. But the mere consideration of the existence of some 
functional laws, whether connected or arbitrary, of dependence of the coefficients m7 0 
on the co-ordinates wv 7’ 2’, or of a B yon x y z, conducts easily, as we have seen, to a 
conical locus of the kind (#"). This result may however be illustrated by the theory 
which we have given of the geometrical relations of the near final ray-lines from an 
oblique plane with the deflected lines of a certain auxiliary paraboloid, and with a 
certain law and constant of deviation. 
For, according to the theory of these relations, the ray-line from a near final point 
B’ on a given oblique plane drawn through the given point B, will or will not inter- 
sect the given final ray-line from B, according as its deviation Sv from its own 
deflected line does or does not compensate for the deviation dy of that deflected line 
from the corresponding plane of deflexure, by these two deviations being equal in 
magnitude but opposite in direction ; the condition of intersection may therefore be 
thus expressed, 
dv + dv=0; (ie) 
or, by the values of the deviations dv, dy, established in the seventeenth number, 
nN = =. sin. 2 + 5. cos. 2, (U") 
that is, 
n (da° + by’) = (t—1) da dy +s (Ox? — by’): (V*) 
and the condition of intersection thus obtained, by the consideration of two equal and 
opposite deviations, is, on account of the meanings (#") (F"") of n, 7, s, ¢, equivalent 
to (Q"), and therefore to the equation (2) of the cone of the second degree. In 
this manner, then, as well as by the former less geometrical process, we might perceive 
that the two planes of vergency for the ray-lines from an oblique plane, (determined 
by (U™) or (V+), and analogous to the two less general planes of vergency consi- 
dered in the sixteenth number,) intersect the oblique plane in the same two lines in 
which that plane intersects a certain cone of the second degree, through the centre of 
which cone it passes ; and that the planes of vergency are imaginary when the oblique 
plane does not intersect this cone. We may remark that the intersection of the 
oblique plane with the cone, or of a near final ray-line from the oblique plane with 
the given final ray-line, is impossible, when the constant of deviation corresponding 
to the oblique plane is greater (abstracting from its sign) than the semidifference of 
the extreme deflexures of the auxiliary paraboloid: for then the compensation of the 
two deviations év, dy, is impossible, the near ray-line always deviating more from the 
corresponding deflected line of the auxiliary paraboloid, than this deflected line from 
