On Systems of Rays. 89 
remark that any two conjugate planes of deflexure, connected by the relation (C), 
intersect the tangent plane of the surface in two conjugate tangents of the kind 
considered by M. Durty. 
Let us now resume the system of ray-lines (C), of which the equations may be 
put by (2) under the form 
=r —z (r0u + sby)—znsy, ) 
(E*) 
y =ty—2 (sox + ty) +2ndx, J 
if we make 
$6 ‘8a 98.08 s 
a 2 (Se am Pie — 155)? ee 
and let us compare these ray-lines with the deflected lines from the auxiliary parabo- 
loid (D"), which have for equations 
vadr—2z (row + soy), y= sy —= (sda + ty). Cw") 
We easily see, by this comparison, that the infinitesimal angle of deviation év of a 
ray-line (EZ) from the corresponding deflected line (J/”™), is still determined by the 
same formula (JZ"°) 
dv=nel, 
as in the simpler theory of the guiding paraboloid explained in the fifteenth number ; 
that is, this angular deviation év is still equal to the perpendicular distance o/ of the 
near final point from the given final ray-line, multiplied by a constant of deviation 7. 
The plane of this angle &v, that is, the plane contaiming the ray-lme (4) and the 
deflected line (JV), has for equation 
vou + y8y = dl? — z (rea* + Qsdxdy + ty"), (G*) 
and therefore contains the right line having for equations 
Bsns eS i 
Pan On rex? + Qsdxby + 18y?? o 
that is, the axis of deflexure (N'”): results which are analogous to those of the 
fifteenth number, expressed by the equations (V™) (O"). And we may construct 
the final ray-line (Z7") by a process of rotation analogous to that already employed, 
namely, by making the deflected line (7), which passes through the two rectan- 
gular axes of deflexure of the auxiliary paraboloid (D"), revolve round the perpen- 
dicular 8/, through the infinitesimal angle 6y, proportional to that perpendicular. The 
theory, therefore, of the guiding paraboloid and constant of deviation, which was 
given in the fifteenth number, for the ray-lines from the near points B, on the final 
perpendicular plane, extends with little modification to the ray-lines from the points 
B on any final oblique plane locus passing through the given final point: namely, 
VOL. XVII. 2A 
