[651
651] ON A SPECIAL SURFACE OF MINIMUM AREA. 61
or, what is the same thing, it is
= ( 2a 2 + 2ab - 2be - 2c 2 )
+ p 2 ( ab- 2ac - be + 2c 2 )
+ pr (— 6a 2 — ab - 14ac + 3b- — 36c )
+ r- (- 8a 2 - 4a6 - 12ac + 36 2 );
so that, writing for convenience a = 1, the equations to be satisfied are
2 — 2c 2 4- 2 (1 — c) 6 = 0,
- 2c + 2c 2 + (1 - c ) 6 = 0,
- 6 - 14c + 36 2 - (1 + 3c) 6 = 0,
— 8 — 12c + 36 2 — 46 = 0.
The first and second are (1-c)(2 + 2c+ 26) = 0 and (1 - c )(-2c+6) = 0 ; viz.
they give c = 1, or else 6 = -f, c = £. In the former case, the third and fourth
equations each become 36 2 -46- 20 = 0, that is (36 — 10) (6 — 2) = 0; in the latter case,
they are satisfied identically; hence we have for a, 6, c the three systems of values
mentioned at the beginning.
This completes the investigation; but it is interesting to find the values assumed
by the other factor of O on substituting therein for a, 6, c the foregoing several
systems of values. We have in general
il = — 2 a 2 q s + 2abq 4 — 2bcq 2 + 2c' 2 q
+ p- (— abq 3 — 2acq 2 + beq + 2c 2 )
-\-pr ( 6a-q 3 — abqf 2 + 14acq — 8b 2 q — 36c)
+ r 2 (— 8a 2 q 2 + 4<abq — 12ac + 36 2 )
= - 2a 2 (q 5 + 1) + 2ab (q 4 - 1) — 26c (q 2 — 1) + 2c 2 (q + 1)
+ p- {— ab (q 3 + 1) — 2ac (q 2 — 1) + be (q + 1)}
+ pr { 6a 2 (q 3 + 1) — a6 (q 2 — 1) + (14ac — 36 2 ) (q + 1)}
+ t 2 { - 8a 2 {q 2 - 1) + 4a6 (q + 1)}
= (q+ 1) j — 2a- (q 4 — q 3 + q' 2 — q + 1) + 2ab (q 3 — q 2 + q - 1) — 26c (<? — 1) + 2c 2 '
+p 2 {- ab(q 2 -q + 1)- 2ac(q- 1) + 6c}
+pr { 6a 2 (q 2 + q + 1)- a6 (q -1) + (14ac - 36 2 )}
q-r 2 { — 8a 2 (q — 1)+ 4a6} >
Hence writing, first, a = c = 1, 6 = we obtain, after some reductions,
O = (q + 1) {- 2q (q - 1) (q 2 - -\°-q + 1) +p 2 (? - l)(-^q-2)+pr(6q 2 - ^q - 10) + r 2 - 8q + ;
secondly, writing a=c=l, 6 = —2, we obtain
il = (q + 1) {- 2 (q + l) 2 (q 2 + 1) +p 2 .2 (q - l) 2 + 2pr (3q 2 -2q + 6)- 8r-q);
and, thirdly, writing a— 1, 6 = —c = —we obtain
O = + 1) {(- 2q 4 + f? 3 - f ? 2 + f q) + p 2 (~ i? 2 + $?“¥) + P r (% 2 —¥ 2 - if) + (“ 8 2 + ¥)}•
