254
ON A SIBI-RECIPROCAL SURFACE.
[671
then the three linear relations satisfied by (a, b, c, f, g, h) would express that the line L
was a line meeting each of the three given lines L x , L 2 , L 3 : the locus is therefore
the quadric surface which passes through these three lines; and I have in my paper
“ On the six coordinates of a Line,” Gamb. Phil. Trans., t. XI. (1869), pp. 290—323,
[435], found the equation to be the foregoing equation T= 0. But it is easy to see that
the same equation subsists in the case where the three equations a 1 f L + b$ x + cji x = 0,
etc., are not satisfied. For the several coefficients being perfectly general, any one of
the three linear relations may be replaced by a linear combination of these equations;
that is, in place of a x , b x , c lf f x , g x , h x , we may write ai, b x ', d, //, gi, hi, where
ai = 0 x a x 4- 0 2 a 2 + 0 3 a 3 , 6/ = 0 x b x + 6 2 b 2 + 6 3 b 3 , etc.; and these factors 6 X , 6 2 , 0 3 may be
conceived to be such that the condition in question a/// + b x 'gi + dW = 0 is satisfied.
Similarly the second set of coefficients may be replaced by ai, b 2 ', c 2 ', fi, gi, hi, where
a 2 = (fi x a x -l - cj) 2 a 2 + <f> 3 a 3 , etc., and the condition aifi + bigi + dhi = 0 is satisfied: and the
third set by a 3 ', b 3 ', d, f 3 ', gi, hi, where a 3 ' = ^r x a x + + ^¡r 3 a 3 , etc., and the condition
aifi + dgi + dhi = 0 is satisfied. We have therefore an equation 0 = (a'g'fi) x* + etc.,
which only differs from the equation T — 0 by having therein the accented letters in
place of the unaccented ones: and, substituting for the accented letters their values,
the whole divides by the determinant and throwing this out we obtain the
required equation T = 0.
But it is easier to obtain the equation T = 0 directly. We have
hy — gz + aw = 0,
— hx . +fz + bw = 0,
gx -fy . +cw = 0,
— ax — by — cz . = 0;
viz. in virtue of the equation af+ bg + ch — 0 which connects the six coordinates, these
four equations are equivalent to two independent equations which are the equations
of the line (a, b, c, f, g, h): or, what is the same thing, any three of these equations
imply the fourth equation and also the relation af+ bg + ch = 0.
We might, from the three linear relations and any three of the last-mentioned
C,f
9> h
and so
obtain
the required
arbitrary
coefficients oc,
A
y, S, to
dimination
is
thus
given
in
the form
w,
-*»
y
= 0,
w,
-
X
w,
-y>
x ,
x ,
y>
fu
ffi,
K
Oi,
b 1}
Cl
u
92,
d,
a 2 ,
b 2 ,
c 2
u
9 3,
h 3 ,
a 3 ,
b 3 >
c 3