6'
II! ! fl«
ON A SPECIAL SURFACE OF MINIMUM AREA.
G — a 2 ( 4g 4 — 16q 2 pr 4- 16gr 2 4- 8p^r 2 )
4- ab (— 6g 3 + 3g 2 /) 2 4- 12qpr — 9r- — 6p 3 r)
4- ac ( 8g 2 — 16g/) 2 + 16pr 4- 4p*)
4- b 2 ( 2 c/ 2 — 4pr)
4- 6c (— 4g 4- 2p 2 ),
Z) = a 2 ( 2g 2 /)?’ — 2 gr 2 — 4/)V)
+ ab (— 4>qpr 4- 2p 3 r)
4- ac (— 4g 2 4- 2g/) 2 - 2/>r)
F =
4- 6 2 (
2pr)
+ be (
2g),
4- Cl 2 (
4g 2 ?’ 2
— 8/)?’ 3 )
4- ab (—
12gr 2
4- 6/) 2 ?’ 2 )
4- ac (—
4g 3
4- 2g 2 /) 2 4- 8qpr — 6?’ 2
4 6 2 (
6r 2 )
4- 6c (
2g 2
- 4/)?’),
a 2 (
4/)?’ 3 )
4- ab (
g 2 /)?’ 4- 3gr 2 — 2/) 2 r 2 )
4- ac (
4g s
— 12g/)?’ 4-12r 2 )
+ 6 2 (
qpr
— 3r 2 )
+ 6c (—
2 g 2
4-
1
45
where in each line the terms are arranged according to their order in p, r
Substituting, we find
)
H =
viz. writing q = — 1, this is
n =
a 2 (—
2g 3 4- 6g 3 /)r
■ -
8 g 2 r 2
4- ab (
2g 4 -
- g 3 /) 2
-
g 2 /)?’ 4-
4- ac (
- 2g 2 /) 2
4- 14g/)?’ —
4- 6 2 (
-
3g/)r 4-
4- 6c (—
2g 2 4- g/) 2
: -
Spr
+ c 2 (
2g -
+■ 2/) 2
a 2 (
2
—
Qpr
- 8r 2 )
4- a6 (
24-
p 2 —
pr
— 4r 2 )
4- ac (
-
2/> 2 — 14/)?'
- 12r 2 )
+ 6 2 (
3/)?’ + 3r 2 )
4- be (—
2 —
/) 2 -
3/)r
)
+ c 2 (-
2 —
2/) 2
)
)
);
so
the
eqt
the
mei
b .y
syst
Hen<
il =
secor
ii =1
and,
0 = (
