344] ON CERTAIN DEVELOPABLE SURFACES, 
.and expressing the determinant in the partially developed form 
269 
= (AM - LH) (FN - CM) 
+ (AN-LG)(FM-BN) 
+ (AP - L 2 )(BG -F 2 ) 
+ (HN- GMf 
+ (HP - LM) (FG - CH) 
+ (GP -LN) (HF-BG), 
and proceeding to the calculation, we find 
AM- LH 
FN- CM 
AN-LG 
FAI-BN 
AP-U 
BC-F 2 
= 3 x 
= 9x 
— 3 x 
= 9 x 
= 3 X 
= 9 x 
acd 2 + 1 
a 2 bd + 1 
abd 2 + 1 
a 2 cd + 1 
a 2 d 2 - 1 
(Pd 1 - 1 
b 2 d 2 + 2 
«V + 4 
ac 2 d - 4 
ab 2 d + 2 
abed + 4 
abcd+ 12 
bc 2 d - 3 
alrc — 7 
b 2 cd — 3 
abc 2 + 1 
b 2 c 2 - 3 
ac 3 — 4 
b 4 +2 
be 3 + 6 
b 3 c - 4 
b 3 d - 4 
b 2 c 2 - 3 
IIN - GM HP-LAI FG-CH GP-LN HF-BG 
= 18 x =3 x = 9 x =3 x =9 x 
ac 3 + 1 
a 2 cd + 1 
abd 2 + 1 
a 2 bd + 1 
acd 2 + 1 
bhl + 1 
ab 2 d- 4 
ac 2 d + 2 
a 2 c 2 + 2 
b 2 d 2 + 4 
b 2 c 2 - 2 
abc 2 — 3 
b 2 cd + 1 
ab 2 c — 3 
bc 2 d - 7 
b 3 c + 6 
be 3 — 4 
c 4 +2 
4. Hence, forming the six parts and collecting, we find 
27 x 
(dd 3 
+ 
1 
+ 
1 
a 3 bcd 3 
— 
12 
+ 
1 
+ 
1 
- 
16 
+ 
1 
+ 
1 
a 3 c 3 d? 
+ 
8 
+ 
4 
- 
4 
+ 
4 
2 
+ 
2 
a 2 b 3 d 3 
+ 
8 
+ 
2 
+ 
2 
+ 
4 
- 
4 
+ 
4 
a 2 b 2 c 2 d 2 
+ 
30 
— 
2 
— 
10 
+ 
54 
- 
10 
— 
2 
d 2 bc 4 d 
_ 
48 
— 
12 
+ 
2 
— 
16 
— 
10 
— 
12 
arc 6 
+ 
16 
+ 
12 
+ 
4 
aided 2 
_ 
48 
— 
12 
— 
10 
- 
16 
+ 
2 
— 
12 
ab 3 c 3 d 
68 
+ 
21 
+ 
25 
- 
48 
+ 
24 
+ 
25 
+ 
21 
ab 2 c 5 
_ 
24 
+ 
6 
+ 
12 
- 
48 
+ 
12 
— 
6 
b 6 d 2 
+ 
16 
+ 
4 
+ 
12 
b r, c‘ 2 d 
_ 
24 
— 
6 
+ 
12 
+ 
12 
- 
48 
+ 
6 
Idc 4 
+ 
9 
— 
24 
+ 
9 
+ 
48 
— 
24