256 
ON THE DIAMETRAL PLANES OF A SURFACE OF THE SECOND ORDER. [39 
A — u, B — u, C — u, so that 
f9L, = @L—(B+ G)u + u 2 , 
33 / = 33 — (O + A) u + u 2 , 
(£/,=(& — (A + B) u + ¿í 2 , 
JF/=3F + Fu > 
n=№ + Hu. 
Hence the equation ax + a!y + a!'z = 0 may be written in the three forms 
& l x + ¥% l y + <& i z = O i 
( &, x + §,y + ^/^ = 0; 
or, what comes to the same thing, as follows, 
f&x + + <Ftz + u {Ax + Hy + Gz) + vx=0, 
Tfyx + 33y + jpz + u {Hx + By + Fz) + vy = 0, 
(§sx + jpy + + u {Gx +Fy + Cz) + vz — 0, 
in which for shortness v has been written instead of 
u 2 — (A + B + C) u. 
The elimination of u, v from these equations gives a result @ = 0, where © is a 
homogeneous function of the third order in x, y, z; and this equation, it is evident, 
must belong to the three diametral planes jointly, i.e. 0 must be the product of three 
linear factors, each of which equated to zero would correspond to a diametral plane. 
Thus the system of diametral planes is given by 
0 = 
@Lx + p^y + Q\jz, Ax + Hy + Gz, 
p^a? + 33y + Jfz, Hx + By+Fz, 
(Six + Jfy + (&z, Gx + Fy + Gz, 
x 
= 0, 
y 
z 
or developing the determinant, as follows, 
® = {G : ffi-H(&)a? + {H$-F r ffi)tf + {F(&-G§)z* 
+ { G (0D - 23) - & (C - B) - (H§ - Fffi)} yz 1 
+ { H {&-(£)-y%(A-C } - {F<&-G§)\zx> 
+ { F №-<&)-§ (B-A)-{G№-H<&))xf 
+ [-H(CD - 33) + (G - B) + (JXffi - G§)} fz 
+ {-F (&-(&)+$ {A-C) + (Crp| - im)} z*x 
+ {- G (33 - m + or (B -A) + {H§-Fffi,} x'-y 
+ ((7^3 - m + - c® + m - H33) x yz -