76] 
SURFACES OF THE THIRD ORDER. 
449 
(!') ~ m ^2n ^ ® + my + ^z + w = 0, 16.25.34 
(“0 
^ x — n -y + n,g + w = 0, 15.24.36 
2Z 
(n ) lx + — y — — 0 + w = 0, 
(p) 
(q) 
(r) 
(p) 
(q) 
(?) 
2m 
'll) 
+ ny + m2 + [mil (p — a) — 21 (1 — m 2 — ?z 2 ) ] = 0, 
p — a 
p + ß 
nx — + lz + [ nl (p — a) — 2m (1 — w 2 — ¿ 2 )] = 6, 
mx + ly — + [ ¿m (p — a) — 2n (1 — Z 2 — m 2 )] ^ ^ = 0, 
2# 1 1 
+ ~y Ar — Z + 
— a n 
m 
1 2y 1 
- x — + jZ + 
n p — a l 
112 z 
-x + jy + 
m l p — a 
1 / \ 2 
— (p- a) - - 
mn L 
w 
m 2 w 2 / J p — /3 
I _ IYI w 
n 2 l 2 ) _ p — ß 
1 , N 2 
lm (p - a) -n 
l 2 m 1 
, * «- a y z , 
(P/) - g" - + - ™~ lmn 
(q,) 
(O 
(P/) - 
w m 
x p — a 2 7 
¿-■V-y+i-fc»» 
Ifl _ - — 1 
m V n 2 l 2 
w 
p-ß 
w 
14.26.35 
51' 
35' 
13.25.46 
= 0, 26' 
= 0, ...16.23.45 
63' 
-«)- 
w 
X V p — CL 
h r - ^ — £m?i 
ml 2 
s( 1 -p-s) (p -“ ) - 
mnj p + ß 
2" 
2' 
Im 
= 0, 
= 0, 
25' 
P + /3 
= 0, ...15.23.46 
P + ß 
0, 
a? + ny + mz - i 2 (1 -m 2 - n 2 )(p-a) - 2mra] = 0, 
Imn 
(q,) + )(p-«)-2^]^ = 0, 36' 
(?,) m# + Zy--^-^2-^[>(l- ¿ 2 — m 2 )(p—a) — 2lm = 0. ...13.26.45 
53' 
61' 
In fact, representing the several functions on the left-hand side of these equations 
respectively by the letters placed opposite to them respectively, the function U is 
expressible in the sixteen forms following 
JJ = wii -f k%yz, 
= wgg + kyzx, 
= whh + k£xy, 
= w60 + k%y £, 
C. 
57