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.

330 MR RICHMOND, ON MINIMAL SURFACES.
ni, m2,... being real numerical constants. The corresponding minimal value of p is of
the form
p = k (n1) - (I + im)k = k +1- (l + im)k + W - (l + ir)+-1,
TW denoting some rational integral homogeneous function of 1, nm, n (with complex numerical
coefficients), of degree k.
It will be seen that of integer values of k the value k = 1 alone fails to give a
function /k (n). It may be easily proved, and will be assumed in what is to come,
that no minimal value of p exists whose denominator is + im and whose numerator
is a rational integral function of the second degree.
9. In order next to determine the most general rational integral function V of
degree k+1 such that the surface
p = V (l + i)k
is minimal, it will be convenient to write for a time
f - l + im, gl-i im,
and to use f, g, n, as independent variables in place of l, m, n. The differential equation
of a minimal surface is now
-L + 4 g = 0,
8n2- a =ag
and is to be satisfied by
p= V -fk
Substituting and multiplying by fk, we find that
aV
g -f + an integral function = 0,
and deduce that the part of V that does not contain f inust consist of a single term,
*C. k+l,
C being a constant. It follows that by subtracting a numerical multiple of the foregoing particular solution we obtain a new minimal function pi, viz.
p = {V- C0 (n)} f k,
in which a factor f is common to numerator and denominator, and may be removed.
By repetition of the argument and process we continually diminish the class of the
surface, and finally establish the theorem:The most general rational mininzmal value of p which has ( +im)k for its denominator is
p = al+ /m + r/3 n 7 +:C (n) + (l + im)8; (s = 2, 3, 4,.../ );
the quantities a,,8, r, C2, C3,... Ck being complex constants.