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.

MR RICHMOND, ON MINIMAL SURFACES. 327
or straight lines, united so as to form a closed contour. Let this contour be enveloped
by a straight line which moves round it, turning always in the same direction; let
the plane of the contour be xOy; let po denote the perpendicular from O on the
enveloping line, and 0b0 the inclination of that perpendicular to Ox.
In a complete circuit of ihe contour, the enveloping line will turn through some
multiple of two right angles, and return to its original position; po is therefore a
periodic function of b0, —the period being a multiple of 7r,-and may be expanded in
a Fourier's series even when p, or its differential coefficients have discontinuities: thus
po = S (ak sin k0o + bk cos kto).
In the case of an oval curve or a closed convex polygon the period of po is 2vr; k
will then receive only integer values. In a cardioid the period is 37r, and 3k will
always l)e an even integer, etc., etc.
The minimal surface sought will be represented by the tangential equation
p = E {(k - cos 0) cot 0 + (k + cos 0) tank 2 0 (ak sin kq + bk cos kc0) + 2k.
For this typical term may be obtained from the general formulae (5) by making
x (u)=-K (tk - u-k), x (y1) =K (tuk - ut-k);
K and K1 being constants suitably chosen; and we may deduce
z = 2 E( k) cotk -tan (ak sin k + bk cos k);
so that, when 0= 7r, z vanishes and p has the correct value.
Interesting special cases arise when the given plane curve is an epicycloid or hypocycloid; for the series for p, then reduces to a single term
p = A cos kdp,
and the required surface is obtained by making in (5)
x (Zt) = B (utk - u-), (It1) = B (ulk - I1l-k).
It is clear however that special surfaces such as this fall under the cases to which
the methods of Darboux are applicable; I therefore pass on to a result which I do
not remember to have seen explicitly stated, (although it follows almost immediately
from several theorems of Darboux), and to some considerations suggested by it. Enough
has been said to shew that integration of Laplace's equation leads rapidly to many of
the chief known results concerning minimal surfaces.
6. Since Laplace's differential equation is linear, the sum of any two of its solutions
is itself a solution: if then pi and p1 be two minimal functions of 1, rm, n, p1i+p is also
a minimal function. Stating this theorem in geometrical language, we enunciate the noteworthy property: