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two smith’s prize dissertations.
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(in the tangent plane at P) of the orthogonal directions PA, PB; viz. it is found
that these must be such that the normals PP', A A' intersect, or, what is the same
thing, the normals PP', BB' (one of these conditions implying the other); that is,
that the lines PA, PB shall be the directions of the two curves of curvature through
P on the given surface.
Observe that PP', A A' intersecting each other, the four points P, P', A, A' are
in the same plane, that is, PA, P'A' intersect, these lines being the normals at P, P'
respectively of the surface through P of one of the other two families; and similarly
PP', BB' intersecting each other, the lines PB, P'B' intersect; these being the normals
at P, P' respectively of the surface through P of the other two families. We have
through PP' two planes at right angles to each other; and these are met by a plane
A'P'B', in two lines A'P', B'P', the inclinations of which to the line PP' differ only
infinitesimally from a right angle, say they are 90° — a and 90° — b respectively; hence
if the angle A'P'B' is = 90° — c, this is the hypotenuse of a right-angled spherical
triangle, the sides whereof are 90° — a, 90° — b; wherefore sin c = sin a sin b, viz. sin c is
an infinitesimal of a higher order which may be neglected, or the angle P' will be
= 90°; that is, the surfaces through P of the other two families, intersecting the given
surface in the directions PA, PB of the two curves of curvature, will intersect the
consecutive surface at P' in the two directions P'A', P'B' at right angles to each
other; which is an a posteriori verification of Dupin’s theorem.
In what precedes the given surface through P may be regarded as a surface
assumed at pleasure; and it in effect appears that taking the consecutive surface also
at pleasure (but varying only infinitesimally from the given surface), the condition in
order that the two surfaces, which at P intersect each other and the given surface at
right angles, shall at P' intersect the consecutive surface in two directions at right
angles to each other, is that they shall intersect the given surface in the directions
PA, PB of the two curves of curvature. But if we thus take the consecutive surface
at pleasure,—or say if we construct it by measuring off along the normal at each
point P of the given surface an infinitesimal distance PP', = p, where p an arbitrary
function of the coordinates of the point P,—then although on the consecutive surface
the lines P'B', P'A' are at right angles to each other, there is nothing to show, and
it is not in fact the case, that these lines P'A', P'B' are the directions of the curves
