Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

326 MR RICHMOND, ON MINIMAL SURFACES.
Although the integration of Laplace's equation presents no difficulty, it is not easy
to say what is the best form of solution of the first degree in the variables which
we should take as the value of p. The formulae due to Weierstrass (D. ~ 188, equations
18), may be obtained from the value
P = r [f (u) +fi/ (u,)] - (i - im)f(u) - (i + im) fj (u,):
but a value which is preferable for the present purpose, in that it is more naturally
attained by integration and leads to simpler results, is
p = r [X (t) + t ' (u1)] - i [X (t) + X1 (ut)]........................ (5);
and this is the value which will be used in the following applications. From it I derive,
by differentiation with respect to 1, m, n, the expressions
= x' (t) +2 1 (1 - uX2) x () + 1< (u1) + 2 i (1 - u12) x 1 (t1);
y = iXy (u) + 2 iu (1 + u2) %" (u) - i1 (ut1)- iut1 (1 + u,12) X1I (1);
z = - X (u) + tx' (t) + t2x" (0) - XI (t1) + UX1' (u.) + u1 "1 (u).
It will be seen that the two forms are in agreement if
f(u) = UX (t); f, (2t) = uXI (ut).
4. As an illustration of the use of these results I consider two methods of solving
the problem of determining a minimal surface which has a given plane as a plane of
symmetry, and cuts that plane at right angles along a given curve; or, as Darboux
(~ 251) expresses it, has a given plane curve as a geodesic. It is clear that if X =X,
(which in the case of a real surface implies that X is a real function), the surface
has zOx as a plane of symmetry and cuts it orthogonally: moreover, if we fix directions
by Euler's two angles, 0 the colatitude and S the longitude, (so that: n: r: sin cos b sin 0 sin b:cos 8: 1,
and u- = e. c-ot 0, ul = e-i. cot ),
2 2
the functions X and XI are determined by the equation
X (cot o ) = Xi (cot o) = - )osec. pd0,
the quantity p being the length of the perpendicular from the origin on any tangent
of the given plane curve, laid in the plane zOx, and 0 the inclination of that perpendicular to Oz.
5. But the following solution is of greater interest, in that it is adapted to cases
when the given plane curve is irregular, being composed of portions of known curves