[650 
bions. 
rsection of 
bs to each 
"2 + W‘ 2 = 0. 
651] 
63 
651. 
ON A SPECIAL SURFACE OF MINIMUM AREA. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xiv. (1877), 
pp. 190—196.] 
A very remarkable form of the surface of minimum area was obtained by Prof. 
Schwarz in his memoir “Bestimmung einer speciellen Minimal-fläche,” Berlin, 1871, 
[Ges. Werke, t. I., pp. 6—125], crowned by the Academy of Sciences at Berlin. The 
equation of the surface is 
1 + pv -¡- v\ + Xp = 0, 
where X, p, v are functions of x, y, z respectively, viz. 
dQ 
-I 
a va^+i p+f)’ 
and y, z are the same functions of p, v respectively. A direct verification of the 
theorem that this is a surface of minimum area, satisfying, that is, the differential 
equation 
r (1 + q 2 ) — 2pqs +1 (1 +p 2 ) = 0, 
is given in the memoir; but the investigation may be conducted in quite a different 
manner, so as to be at once symmetrical and somewhat more general, viz. we may 
enquire whether there exists a surface of minimum area 
1 + fiv + v\ + X/4 — 0, 
where the determining equations are 
V 2 = a A, 4 + b\ 2 + c, 
p' 2 = a/i 4 + bp 2 + c, 
v 1 '- = av* + bv 2 + c,