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.

XVI. On Minzinmal Su faces. By H. W. RIICHIMOND, M.A., King's College,
Cambridge.
[Received 10 November 1899.]
1. IN a short paper read before the London Mathematical Society on Feb. 9 last,
and since printed in the Proceedings of the Society, Vol. xxx. p. 276, Mr T. J. I'A.
Bromwich has noted an interesting form of the tangential equation of a minimal surface,
by which the determination of such surfaces is made to depend upon a particular type
of solution of Laplace's equation. The idea of thus establishing a connexion between
certain of Laplace's functions and minimal surfaces is one that presented itself to me
several years ago, and led me then (in 1891-92) to consider at some length to what
extent the study of these surfaces given by Darboux in Part I, Book III, of his
Théorie générale des Szrfaces might be modified by this connexion. Although the
familiar treatment of Laplace's equation led me, (in many instances by simpler paths
than Darboux), to a number of the chief known theorems concerning minimal surfaces,
yet I never succeeded in reaching untrodden ground, and for this reason laid aside
iny work; but the appearance of Mr Brornwich's paper has caused me to look through
my notes, and to consider with some fulness a special family of algebraic minimal
surfaces to which the method is peculiarly applicable.
So thorough a discussion of the history and properties of minimal surfaces is given
by Darboux, in Book III. of his Théorie générale des Surfaces, that it will seldom be
necessary to refer to other sources of information: references to Darboux will be made
simply by the letter D. followed by the number of the paragraph in question;-thus
(D. ~ 175). In all that follows it is supposed that a system of real rectangular
Cartesian axes is employed.
2. The tangential equation of a surface,
(p. 1), 2., n)= 0,
(where b is a homoyeneous function of p, 1, n, n but not necessarily algebraic), expresses the condition that the plane
x + y + nz = p......................................... (1),
should be tangent to the surface. Should / be rational, integral and homogeneous of
the kth degree, the surface is algebraic and of the kth class.