90 Professor Hamitton’s Third Supplement 
by employing a more general auxiliary paraboloid, and by considering deflexures and 
deflected lines, instead of curvatures and normals. And we may transfer to this more 
general auxiliary paraboloid, and to its connected constant of deviation, the reason- 
ings of the sixteenth number, respecting the system of final ray-lines ; for example, 
the reasonings respecting the foci by projection, and those respecting the condition of 
intersection of such ray-lmes. And since for any given values of p, g, that is, for 
any given position of the oblique plane ("), we can construct the new auxiliary 
paraboloid (D"), and its new constant of deviation (#'"), by the coefficients 
Sa 38 3a 38 & 3B 
dz’ 82? By” dy” Sz’ Se’ 
that is, by means of the former guiding paraboloid (Z°) and the former constant of 
deviation (B"), and by the magnitude and plane of curvature (7"°) of the final ray, 
we may be considered as having reduced the theory of the geometrical arrangement 
and relations of the system of final ray-lines (C™), from an oblique plane (B"), to 
the theory of the elements of arrangement, which was given in the fifteenth number. 
Construction of the New Auxiliary Paraboloid, (or of an Osculating Hyperboloid, ) 
and of the New Constant of Deviation, for Ray-lines from an Oblique Plane, 
by the former Elements of Arrangement. 
18. To construct the new auxiliary paraboloid (D") by the former elements of 
arrangement, we may observe that this new paraboloid not only touches the given 
oblique plane (6) at the given final point B of the original luminous path, but 
osculates in all directions at that given point to a certain hyperboloid, represented by 
the following equation, 
2=pr+qyttret+sjaythty—tz ( we y =) 5 ae) 
in which 7, s, ¢, are the particular values 
peo Coe boy 28 #0 8B 13 
Urs ox’ $= 2 ate t,= by’ (K ) 
of the coefficients r s ¢, deduced from the general expressions (#*) by making 
D=O;9=9; (L*) 
that is, by passing to the case of no obliquity ; so that the equation (Z*) of the guiding 
paraboloid may be put under the form 
z=trgets.acyttty’s (M”) 
which includes the form (G"). Reciprocally, the sought paraboloid (D") is the only 
paraboloid which has its indefinite axis parallel to the given final ray-line, and oscu- 
