88 Professor Hamitton’s Third Supplement 
Before we proceed to apply these general remarks on the deflexures of surfaces to 
the optical question proposed in the present number, that is, to the study of the con- 
nexion of the ray-lines (C'”) with the paraboloid (D”), we may remark that the 
theory which M. Durty has given, in his excellent Développements de Géométrie, of 
the indicating curves and conjugate tangents of a surface, may be extended from cur- 
vatures to deflexures. For if we consider the deflexion (402=4f0l*) in the given 
arbitrary direction of z as equal to any given infinitesimal quantity of the second 
order, that is, if we cut the given surface by a plane ~ 
2—px —qy =4,%z=deflexion = const. , (A®) 
parallel and infinitely near to the given tangent plane (#""), we obtain in general a 
plane curve of section which may be considered as of the second degree, namely, the 
indicating curve considered by M. Durty, of which the axes by their directions and 
values indicate the shape of the given surface near the given point, by indicating its 
curvatures and planes of curvature. This indicating curve is on the following cylin- 
der of the second degree, which has for its indefinite axis the axis of deflexion, and 
which we shall call the indicating cylinder of deflexion, 
ra? + Qsay + ty? =z = const. ; (B") 
and it is easy to see that the two principal planes of deflexure, ¢,, ¢., are the princi- 
pal diametral planes of this indicating cylinder, and that the two principal deflexures 
Sis fz» positive or negative, are equal respectively to the given double deflexion 
ez divided by the squares of the real or imaginary principal semidiameters or 
semiaxes of the cylinder, perpendicular to its indefinite axis. In general, the 
positive or negative deflexure f; corresponding to any plane of deflexure ¢, is equal 
to the given double deflexion 6’z divided by the square of the real or imaginary semi- 
diameter of the cylinder, contained in this plane of deflexure, and perpendicular to 
the axis of deflexion, that is, to the indefinite axis of the cylinder. Hence it follows, 
that if we consider any two conjugate diametral planes ¢, ¢,, which we shall call con- 
sugate planes of deflecure, and which are connected by the relation 
O=r+s(tan. ¢+tan. ¢) +¢. tan. » tan. ¢,, (€*) 
the sum of the two corresponding conjugate radii of deflexure, ; + Z , is constant, and 
equal to the sum of the two extreme or principal radii : that is, we have 
UN eal 1 1 
at fhe | 
a relation which might also have been deduced from the general expression for 
the deflexure, without its being necessary to employ the indicating cylinder. We may 
(D") 
we 
