76] 
SURFACES OF THE THIRD ORDER. 
447 
m 
I have found that by expressing the parameter k in the particular form 
p 2 — (imn — j—■—''j 
V Imn) 
k=- 
2 [p — Imn 
Imnj 
or, as this equation may be more conveniently written, 
p 2 — B 2 1 1 
k = -x; a = Imn + ^— , /3 = Imn — 7 — , t 1 ) 
2(p — a) Imn Imn v 7 
the equations of all the planes are expressible in a rational form. These equations are 
in fact the following: [I have added, here and in the table p. 450, the reference 
numbers 12', 23', &c. constituting a different notation for the lines and planes.] 
(w) w=0 12' 
(3) 
lx + my + nz + w 
x y z 
y q 1— + w 
Imn 
x = 0, 12.34.56 
y = 0, 42' 
z= 0, 14' 
x + (n — w = 0, 
k\ m V n) 
T2 
1)““ 0 2 ' 3 
lx + — + - + w = 0, 
m n 
t + my + - + w = 0, 
l n 
T +2- +nz + w = 0, 13.24.56 
L m 
1 A somewhat more elegant form is obtained by writing p = 2q + a\ this gives 
k= \- + lmn > ( i+ i) -*=•