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.

328 MR RICHMOND, ON MINIMAL SURFACES.
If any two minimal surfaces be taken, the locus of the middle points of lines which
join the points of contact of parallel tangent planes is also a minimal surface.
But, conversely, the possibility that a given minimal value of p may be resolved into
the sum of two or more simpler values is suggested by the theorem. I propose to carry
through this idea in the case of rational algebraic minimal functions;-to prove that every
rational algebraic minimal function may be expressed as the sum of a finite number of
such functions each belonging to certain standard types, much in the way that every
rational fraction may be broken into partial fractions. In other words, I hope to establish
that by taking a finite number of minimal surfaces of certain normal types, disposed in
space with various orientations, and constructing the locus of the centre of mean position
of the points of contact of parallel tangent planes, we may arrive at any minimal surface
whatever, for which p is a rational algebraic homogeneous function of 1, m, n, of the
first degree.
When p is such a function, the surface, whether minimal or not, will have one and
only one tangent plane parallel to any given plane: if the surface be of class k + 1 it will
have the plane infinity as a k-fold tangent plane, and must therefore be reciprocal to
what Cayley called a Monoid surface: (Comptes Rendus, t. 54, 1862, pp. 55, 396, 672). A
paraboloid is the simplest instance of the surfaces we are considering. Now the analogous
curves in plane geometry presented themselves to Clifford's notice in the course of' that
wonderful chain of reasoning, the Synthetic Proof of Miquel's Theorem, (Collected Works,
p. 38), and were named by him double, triple,... k-fold, parcabolas. Following his example,
1 call a surface of class k +l, which has the plane infinity as a k-fold plane, a k-fold paraboloid; and the family of such surfaces, (the value of k not being specified), Multiple
Paraboloids.
7. The tangential equation of a k-fold paraboloid will be written as
p=PV U,
U and V being rational integral homogeneous functions of 1, m, n, of degree k and k + 1
respectively. If for the moment partial differentiations with regard to 1, m, n, be indicated
by suffixes 1, 2, 3, respectively, the condition (4) that the surface should be minimal gives
us the identity
V ( ll + U22 + U33)- U(Vll + V22 +V33)+ 2 (U1 U2V+ U2 +U3V3)-2 V(U12 + U22+ U32) U;
and so proves that ( U12 + U22 + U32) U
is a rational integral function of 1, m, n:-a result possible only if U be the product of
factors which are powers either of
12 + q2 + n22,
or of linear functions such as al + bm + cn,
in which a2 + b2 + c2 = O.