The collected mathematical papers of Arthur Cayley.

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A DEMONSTRATION OF DUPIN S THEOREM.

89

Let the intersection be a curve of curvature on the first surface; the successive
normals intersect, giving rise to a developable, and the intersection of the two surfaces,
say I, is an involute of the edge of regression of this developable, say of the curve C.
The successive normals of the second surface are the lines at the different points of
I at right angles to the planes of the developable, that is, to the osculating planes
of C; or, what is the same thing, they are lines parallel to the binormals of C (the
line at any point of a curve, at right angles to the osculating plane, is termed the
"binormal"). But if the intersection I is a curve of curvature on the second surface,
then the successive lines intersect; that is, starting from the curve C, the theorem in
effect is that at each point of the involute drawing a line parallel to the binormal
of the corresponding point of the curve, the successive lines intersect, giving rise to a
developable. To prove this, let the arc s be measured from any fixed point of the
curve, and the coordinates x, y, z be considered as functions of s; and let xa, x', x"'
dx d2x d3x
denote d 2 ds 3 and the like as regards y and z. Measuring off on the tangent
at the point (x, y, z) a length I-s, the locus of the extremity is the involute;
that is, for the point (x, y, z) on the curve, the coordinates of the corresponding point on
the involute are x 4 (1 - s) x', y + (I - s) y', z + ( - s) z'. Moreover, the cosine inclinations
of the binormal are as y'z" - y"z', z'x"- z"x', 'y" - xy'. Hence taking X, Y, Z as
current coordinates, the equations of the line parallel to the binormal may be written
X = + (I -s) ' + (y'z" -y"z'),
Y = y + ( - s) y' + 6 (z'x" - "x'),
Z = + (- s) ' + O ('Y - y "y'),
and the condition of intersection is therefore
x", y'z - y"z', (y'" - y"z')' =0.
y", z'x" - z"I', (z'x" - z')'
z", x'y"- x"y', (X'y"- /"y')'
Form a minor out of the first and second columns, e.g.
y" (x' y" - x"y) - z" (z'X" - Z"x'),
this is,
xC2 (x"/2 + y + 2) ( + Y + y2), + z) x ( yy +
or the last term being = 0, and the factor x"2 + y"2+ z"2 being common, the minors
are as x': y': z'. Moreover (y'z" - y"z')'= y'z' - y"'z', &c., hence the determinant is
XI (y'Z'" - y 'z) + y' ('x" - Z'cx') + z' (x'y'" - x'),
viz. this is =0, or the theorem is proved.

C. IX.

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