27 4
ON CERTAIN DEVELOPABLE SURFACES.
[344
L 2 = 9 x
3P = 64 x
JV 2 = 96 x
2MN =96 x
2NL — 36 x
2LM = 48 x
a 4 e 2 + 1
cc 3 c 2 e + 4
a 2 b 2 ce — 8
a 2 c 4 + 4
ab 2 c 3 — 16
6 4 c 2 + 16
co 2 b 2 c 2 + 9
ctb 4 c - 24
b 6 + 16
a 4 c 2 + 1
cvPc - 4
a 2 b 4 + 4
d 3 bc 2 — 3
cc 2 b 3 c + 10
cob 5 - 8
a 4 ce + 1
a :i b‘ 2 e - 2
a 3 c 3 + 2
ar/rc 2 — 8
ab 4 c + 8
o?bce — 3
a 2 b s e + 4
a 2 bc 3 — 6
ab 3 c 2 +20
Vc - 16
13. It is now easy to form the seven component terms of the determinant, and
thence the determinant itself; each of the component terms divides by 576, and
omitting this factor, the sum of the seven terms divides by 2; the result is
co 7 ce 4
+
3
3
+
9
a 6 c 3 e 2
+
28
—
10
+
120
+
18
—
72
a 5 b 2 c 2 e 3
—
96
+
24
—
360
—
72
—
144
+
144
4-
360
—
144
a 5 c 5 e 2
+
90
+
15
+
399
+
42
_
276
a 4 b 4 ce 3
+
24
+
72
+
144
+
192
4-
360
—
480
-
720
4-
480
a 4 b 2 c 4 e 2
—
384
—
168
-
1944
—
216
_
168
_
144
4-
1584
+
288
co 4 c 7 e
+
108
+
60
4*
444
+
12
—
300
a 3 b 6 e 3
+
32
—
96
—
128
_
288
+
384
+
576
_
384
a 3 b 4 c 3 e 2
+
568
+
200
+
3024
—
288
—
168
+
1056
+
3504
+
480
a 3 b 2 c 6 e
—
224
—
112
—
2616
4“
432
_
48
—
576
+
1752
+
720
a 3 c 9
+
27
4-
18
+
108
—
72
a 2 b 6 c 2 e 2
+
384
—
1152
+
2496
2304
+
4032
—
2304
a 2 b 4 c 5 e
—
696
+
6336
—
2304
4*
48
+
1920
—
3552
—
3840
aW
192
—
336
+
1296
+
288
—
864
ab s ce 2
—
1152
—
3072
+
1536
_
2304
+
1536
ab e c 4 e
+
1440
—
6912
+
3840
—
1536
+
2496
+
4992
ab 4 c 7
—
408
+
48
—
3456
_
288
+
2880
b 10 e 2
512
+
1024
b s c 3 e
—
640
+
2304
—
2048
_
1536
b s c 6
+
192
+
384
+
2304
-
2304
where the first column is the Hessian.
14. This divides as it should do by
U =
and the quotient, which is the Prohessian, is found to be
PU =
a 3 e 2
+ 1
a 2 c 2 e
+ 6
ab 2 ce
- 24
ac 4
+ 9
b 4 e
+ 16
b 2 c 3
- 8
sian, is
found
a 4 ce 2
+ 3
co 3 c 3 e
+ 10
a 2 b 2 c 2 e
- 24
a 2 c 5
+ 3
ccb 4 ce
- 24
ab 2 c 4
+ 24
b 6 e
+ 32
b 4 c 3
- 24
