[344 
344] 
ON CERTAIN DEVELOPABLE SURFACES. 
279 
Special Sextic Developable, Nos. 26 to 35. 
26. Lastly, we have the developable 
TJ = cv'e’ j — \2a 2 bde 2 — 27a 2 d 4 — 6ab 2 d 2 e - 27b 4 e 2 — 64b 3 d 3 — 0, 
lr| d of derived from the quartic function (a, b, 0, d, e^t, l) 4 ; the discriminant is in fact 
(ae — 4bdf — 27 (— ad 2 — b 2 e) 2 , which is equal to the foregoing value of U. 
, 27. Taking a, b, d, e as coordinates, then omitting common numerical factors, the 
first derived functions are 
a 2 e 3 — 8 abde 2 — 18ad 4 — 2 b 2 d 2 e, 
— 4a 2 de 2 — 4<abd 2 e— 3Qb 3 e 2 — 64<b 2 d 3 , 
— 4a 2 be 2 — SQa 2 d 3 — 4<ab 2 de — 64<b 3 d 2 , 
a 3 e 2 — 8arbde— 2ab 2 d 2 — 18b 4 e , 
and the second derived functions (changing, for greater convenience, the signs) are 
2 (— ae 3 + 4tbde 2 + Dd 4 ), 4(2ad 2 + bd 2 e), 4> (2abe 2 + ISad 3 + b 2 de), — Sa 2 e 2 + 16abde + 2b 2 d 2 , 
4 (2ade 2 + bd 2 e), 4 (ad 2 e + 27b 2 e 2 + 32bd s ), 4 (a 2 e 2 + 2abde + 4t8b 2 d 2 ), 4 (2a 2 de + abd 1 + 186 3 e), 
4 (2abe 2 + 18aci 3 + b 2 de), 4 (a 2 e 2 + 2abde + 4<8b 2 d 2 ), 4 (27a 2 d 2 + ab 2 e + 2>2b 3 d), 4 (2cC-be + ab 2 d), 
— 3a 2 e 2 + 16abde + 2b 2 d 2 , 4 (2a 2 de 4- abd 2 + 18b 3 e), 4 (2a 2 be + ab 2 d), 2 (— a?e + 4a 2 bd + Ob 4 )- 
28. Representing these by 
A > 
H, 
G, 
L, 
H, 
B, 
F, 
M, 
G , 
F, 
G, 
M, 
L, 
M, 
N, 
F, 
and employing for the determinant the same partially developed form as in the first 
example, then proceeding to the calculation, we find 
AM-LU 
= 4 x 
FN-CM 
= 16 X 
AN- LG 
= 4 x 
FM- BN 
= 16 X 
AP-L 2 
= 1 X 
BC -F 2 
= 16 X 
— 
1IN-GM 
= 288 x 
s 
cdde 4 + 2 
a 2 bd 2 e 3 — 15 
a 2 d 5 e + 36 
ab 3 e 4 — 36 
abW- 12 
abd 6 + 18 
b 4 de 3 +144 
b 3 d 4 e + 322 
\ 4 bd + 2 
a i d i e - 54 
a?b 2 de 2 + 3 
d 3 bd 4 - 27 
aWe- 453 
aFe 2 - 18 
ab 4 d 3 + 16 
b 6 de -576 
a :t be 4 + 2 
a 3 d 3 e 2 + 54 
a 2 b 2 de?~ 15 
a 2 bd 4 e - 252 
ab 3 d 2 e 2 — 12 
ab 2 d 5 - 18 
lAPe - 2 
aW + 2' 
d 3 bd 2 e 2 + 3 
d 2 b 3 e 3 — 36 
a 2 b 2 d 3 e + 33 
ab 4 de 2 + 9 
add 4 + 16 
Fd 2 e + 864 
a 4 e 4 — 5 
d 2 bdd + 64 
d’d 4 e — 36 
a 2 b 2 d 2 e 2 —180 
d 2 bd 5 +144 
ab 4 e 3 — 36 
ab 3 d 3 e — 64 
If’de 2 +144 
b 4 d 4 + 320 
« 4 e 4 — 1 
a 3 bde 3 — 4 
a 3 d 4 e + 27 
a 2 b 2 d 2 e 2 + 630 
add 5 +864 
ab 4 e 3 + 27 
abhl 3 e - 128 
¥de 2 + 864 
b 4 d 4 -1280 
d'd 4 e — 2 
d 2 bd? — 1 
ab 4 e 3 — 2 
ab 3 d 3 e-\8 
b 5 de 2 — 1 
+ 459 -1107 - 243 +891 + 351 -999 - 24