344] 
ON CERTAIN DEVELOPABLE SURFACES. 
273 
and writing therein in place of (a, b, c, e), the values (a, 2b, 3c, — 27e), the second 
factor divided by 729 gives the foregoing expression for TJ, belonging to the form 
(a, 2b, Sc, 0, -27e#t, l) 4 . 
11. Taking (a, b, c, e) as coordinates, and omitting common numerical factors, the 
first derived functions of U are 
3a 2 e 2 + 12ac 2 e — 24b 2 ce + 9c 4 , 
— 48abce + 64b 3 e — 16bc s , 
12(1? ce — 24ab 2 e + 36ac 3 — 24 b 2 c 2 , 
2a 3 e + 6a 2 c 2 — 24aZ> 2 c + 166 4 , 
and the second derived functions are 
3 (ae 2 + 2c 2 e) , — 24 bee , 6 (2 ace — 2 b 2 e + 3c 3 ), 
— 24bee , 8 (— 3ace + 12b 2 e — c 3 ), 24 (— abe — be 2 ) , 
6 (2ace — 2b 2 e + Sc 3 ), 24 (— abe — be 2 ) , 6 (a 2 e + 9ac 2 — 4b 2 c), 
3 (a 2 e + 2ac 2 — 4b 2 c), 8 (— 3abe + 4b 3 ) , 6 (a 2 c — 2ab 2 ) , 
3 (a 2 e + 2ac 2 — 4 b 2 c), 
8 (— 3 abe + 4b 3 ) , 
6 (a 2 c — 2a6 2 ) , 
a s 
12. Kepresenting these by 
A, 
H, 
G, 
L, 
H, 
B, 
F, 
M, 
G, 
F, 
c, 
N, 
L, 
M, 
N, 
P, 
and expressing the determinant in the partially developed form 
P (ABC- AF 2 - BG 2 - CH 2 + 2FGH) 
- D (BC - F 2 ) - M 2 (CA - G 2 ) - N 2 (AB - H 2 ) 
- 2 MN(GH - AF) - 2NL (HF- BG) - 2LM(FG - CH), 
then, proceeding to the calculation, we have 
BC - F 2 
— 48 x 
CA - G 2 
= 18 x 
AB-H 2 
= 24 x . 
G11 — A F 
= 72 x 
HF-BG 
= 48 x 
FG-CH 
= 144 x 
ABC-à c. 
= 288 x 
a 3 ce 2 — 3 
a 2 c 3 e - 28 
ab 2 c 2 e +96 
ac 5 — 9 
b 4 ce — 48 
b 2 c 4 - 8 
a 3 e 2 + 1 
a 2 c 2 e 2 + 3 
ab 2 ce 2 + 12 
ac*e — 6 
b 4 e 2 - 8 
b 2 c 3 e + 16 
c G - 18 
a 2 ce 3 — 3 
ab 2 e 3 + 12 
ac 3 e 2 — 7 
c r, e — 2 
a 2 be 3 + 1 
abc 2 e 2 - 1 
b 3 ce 2 + 4 
bc 4 e — 4 
tree 2 + 6 
ab 2 ce 2 - 18 
ac 4 e +11 
b 4 e 2 + 24 
b 2 c 3 e — 26 
c B +3 
a 2 bcc 2 - 1 
ab 3 c 2 + 2 
abc 3 e + 4 
b 3 c 2 e — 2 
be 5 - 3 
-N 
alee 4 — 3 
a 3 c 3 e 3 — 10 
a 2 b 2 c 2 e 3 + 24 
a 2 c 5 e 2 + 15 
ab 4 ce 3 + 72 1 
ab 2 c 4 e 2 - 168 
ac 7 e + 60 
b 6 e 3 — 96 
b 4 c 3 e 2 -200 
b 3 c 6 e -112 
c 9 + 18 
c. v. 3a