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. 331
10. The same method is applicable to the case when the denominator U contains
other factors besides f: for if we substitute
p= V (fh.S)
in the differential equation we find on multiplying by fh that, if S do not contain f
as a factor,
avr as
ag ag
must be divisible by f, and infer that the terms in V that are independent of f must be
equal to those in S multiplied by nh+1 and a constant. If h be equal to unity we must
therefore have
V= A. n2. S+ terms divisible by f;
but substitution in the differential equation proves that A must vanish: if on the other
hand h be greater than unity, we may, by subtracting a properly chosen multiple of
h (n) +fh,
obtain a new minimal function whose denominator does not contain so high a power of f
as fh. It follows that the most general rational minimal function with denominator
( + im)h. S
may be obtained by adding to a value with denominator S the terms
C,. s (n). (1 + im)s: (s = 2, 3, 4,... h):
C2, C,... Ch, being complex constants.
The factors of S may now be subjected to the same treatment; that is to say,
first reduced to the form 1 + im by a real transformation of axes, and then made to
yield a series of fractions of the types already discovered. The most general minimal
value of p which is a rational function of 1, m, n, may therefore be resolved into the
sum of a number of terms each separately capable of being reduced by a real transformation of axes to one of the types already quoted.
11. The simplest value of p of the kind we are considering is obtained when
k= 2, viz.
2p (1 + im)2 = 2n3 + 3n (12 + m2),
and leads to a surface,
2 (x + iy)= = 18 (x + iy) + 27 (x- iy),
of class and order three: but, as imaginary surfaces such as this are of minor interest,
we may pass on to the discussion of the case when the surface is real.
In order that the surface should be real, each of the typical complex terms into
which p was broken up must be accompanied by the conjugate complex term, the
numerical constants multiplying each also being conjugate imaginaries: a rotation of the
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