86 Professor Hamitton’s Third Supplement 
We may also give the name of ais of deflexure, to the axis of this circle, that is, to 
the right line haying for equations 
y= —x cotan. 9, — : (N") 
and we easily see that there are two principal circles of deflexure, analogous to the 
two principal circles of curvature, namely, the two circles having for equations 
Firs, y=0, a+ 2° aah 
wed (O”) 
Second a=0, y+ 2=—>3 
fr 
and two principal rectangular axes of deflexure, namely, 
First @=0; 2 == ; Second y=0, z= - : (Gis) 
1 
These principal axes of deflexure are analogous to the principal axes of curvature, 
that is, to the axes of the two principal osculating circles of the normal sections, in 
the less general theory of normals. And as, in that theory, the near normals all pass 
through the two principal axes of curvature, so we may now consider a more general 
system of right lines, which we shall call the deflected lines, all near the arbitrary axis 
of deflexion, and all passing through the two corresponding principal axes of deflexure, 
and therefore haying for equations, 
wade de, y=y—-fiy, — (Q”) 
when the co-ordinates are chosen as before. These deflected lines are normals, 
in the present order of approximation, to the locus of the circles of deflexure (7 2), 
that is, to the surface of the fourth degree 
: a 22 (a7) ' 
2 Dita ud 12 
P+yPr+e= Fitifey ? (R”) 
and they might be defined by this condition, or by the condition that they are nor- 
mals, in the same order of approximation, to the following paraboloid, 
c=i(fietfiy’)» (s") 
which osculates to the locus (#"), and has the property that its ordinates measure 
the deflexions (G") of the given surface. 
A deflected line of the system (@”) is in the corresponding plane of deflexure 
YOu = LOY, Ci) 
if that plane coincide with either of those two principal rectangular planes of deflex- 
ure, which we have taken for co-ordinate planes; but otherwise the deflected line 
makes with the plane of deflexure an infinitesimal angle oy, expressed as follows, 
&=2( fi-—fr) ol. sin. 2¢: (U") 
