64 
ON A SPECIAL SURFACE OF MINIMUM AREA. 
[651 
\ = dec ’ ^ ^at coefficiente a > c must satisfy four homogeneous 
quadric equations, which, in fact, admit of simultaneous solution, and that in three 
distinct ways; viz. assuming a = 1, the solutions are 
a — 1, 6= c= 1, 
a = 1, b = — 2 , c= 1, 
u = l, b — ^ , c — — -g-j 
that is, 
X' 2 = A 4 + ^-X 2 + 1 {= | (£X 4 + f X 2 + £)}, 
which gives Schwarz’s surface: 
A' 2 = A 4 — 2A 2 + 1 or A'= ± (A 2 - 1), 
which, it is easy to see, gives only x + y + z = const.; and 
=(x’-i)(v+i), 
which is a surface similar in its nature to Schwarz’s surface. 
The investigation is as follows: the condition to be satisfied by a surface of 
minimum area TJ = 0 is 
(a + b + c) (X 2 + F 2 + Z 2 ) — (a, b, c, f, g, h#X, Y, Zf = 0, 
where (X, Y, Z) are the first derived coefficients and (a, b, c, f, g, h) the second 
derived coefficients of TJ in regard to the coordinates. Considering TJ as a function 
of A, g, v, which are functions of x, y, z respectively, and writing (X, M, X) and 
(a, b, c, f, g, h) for the first and second derived functions of U in regard to A, g, v, 
also A', A" for the first and second derived functions of A in regard to x, and so 
for g, g" and v, v": we have 
(X, F, Z) = (XA', Mg', Nv'), 
(a, b, c, f, g, h) = (aX' 2 + XA", by 2 + Mg", cv 2 + Nv", fg'v', gv A', hX'g), 
and for the particular surface U = 1 + gv + vX + Xg = 0, the values are 
(X, M, N, a, b, c, f, g, h) = (g + v, v + X, A + g, 0, 0, 0, 1, 1, 1). 
Hence the condition is found to be 
2g' 2 v' 2 (A + g) (A + v) 
+ 2z/ 2 A' 2 (g + v) (g + A) 
+ 2A ~g 2 (y + A) (v + g) 
— X" (g + v) {(A + v) 2 g 2 + (A + g) 2 v' 2 } 
— g" {v + A) {(g + A) 2 v 2 + (g + v) 2 A' 2 } 
— v" (A + g) {(v 4- g) 2 A' 2 + (v + A) 2 g' 2 ) — 0,