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