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. 325
The equation 0b (p, 1, m, n)= 0 will always be regarded as defining a dependent
variable p as a function of three independent variables 1, m, n;p =,(,, )............................................(2);
but the function r is of necessity homogeneous and of the first degree. The coordinates of the point of contact of the plane (1) with the surface enveloped by it are
ap. =3 a1 ap
ap; Y=aP'.......
so that x, y, z are expressed as homogeneous functions of 1, mz, n, of degree zero, i.e.
as functions of the ratios: m:n. It is therefore possible to eliminate 1, m, n from
equations (3) and so to obtain a relation in x, y, z, alone, the equation of the surface
in point coordinates. The condition that the surface should be minimal is established
without difficulty, viz.
ap+ p ap (4)
- ',+ -. =o..........................................4
Hence:-When p is a function of 1, m, n, homogeneous and of the first degree,
which satisfies Laplace's equation, the envelop of the planes (1), or the locus of the point
(3), is a minimal surface. When the condition (4) is satisfied, I shall say that p has
a minimal value, or is a minimal function of l, m, n.
It is of importance to observe that, in what precedes, the condition
12 +,2 + n2 = 1,
is not imposed: provided only that p is of the first degree in 1, m, n (which is
always to be understood in future), it is absolutely immaterial whether the sum of the
squares of these quantities be equal to unity or no. When (4) is satisfied it is easy
to establish the theorem of M. Ossian Bonnet (D. ~~ 202, 203), that the horograph of
a minimal surface is a conformable map of the surface.
3. I now consider very briefly to what results the common manipulation of Laplace's
equation leads. Since p satisfies the equation, so also do its three partial differential
coefficients, which, as we have seen, are the coordinates of points of the surface, expressed
in terms of the ratios 1: m: n. Now the solutions of Laplace's equation which are of
degree zero in the variables are of the form,
F (v) + F1 (ut),
~where l~i r+ + it- r + zn - im? r + n
where u=- =. u.= =
r- - tim' r - n I + in'
and r = (12 + m2 + n2).
These quantities u and u1 are thus the same as those of Darboux (cf. D. ~~ 193,
195). The formulae of Weierstrass (D. ~ 188, equation 17) are readily deduced; while
if we take new variables v and vy, the former a function of u and the latter of u1,
we reach the solution of Monge (1D. ~~ 179 and 218).