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553] two smith’s prize dissertations.
then the requisite conditions are
dF dA> dF <№> dF d<$ _ Q
dx dx dy dy dz dz ’
dF cN' _ n
dx dx ’
d<& dW _ A
dx dx ’
viz. these equations must be satisfied, not in general identically, but in virtue of the
given equations F= 0, <f> = 0, = 0.
Or, what is more convenient, we may take the equations of the three families
to be
y> Z ) = Q, q-<f>(x, y, z) = 0, r - y}r (x, y, z) = 0;
and write the conditions in the form
dp dq ^ dp dq dp dq _
dx dx dy dy dz dz ’
dp dr A
dx dx ’
dq dr A
dx dx ’
where of course p, q, r stand for their given functional values, p = f(x, y, z), &c.; the
equations in this form contain only (x, y, z), and not the parameters p, q, r\ so that,
if satisfied at all, they must be satisfied identically; and the required conditions therefore
are that the last-mentioned system of equations shall be satisfied identically by the
values p, q, r considered as given functions of (x, y, z).
The last-mentioned conditions lead to the theorem known as Dupin’s; viz. it
follows from them that the surfaces intersect along their curves of curvature; or more
definitely, each surface of one family is intersected by the surfaces of the other two
families in its two sets of curves of curvature respectively.
To indicate the geometrical ground of the theorem, consider on a surface of one
family a point P, and at this point the normal meeting the consecutive surface in P';
the surfaces through P of the other two families respectively will pass through P',
and meet the given surface in two curves PA, PB (viz. PA, PB represent infinitesimal
arcs on these two curves respectively), the angle at P being a right angle.
Drawing at A, B normals to the given surface to meet the consecutive surface
in the points A', B' respectively, the same two surfaces will meet the consecutive
surface in the arcs P'A', P'B' respectively; and (the system being orthogonal), we
must have the angle at P' a right angle. This imposes a condition upon the direction
c. vm. 71
