80 
ON THE SIX COORDINATES OF A LINE. 
[435 
but also that af + bg x + chi + fa x + gb x + c/q = 0 and af, + bg 2 + ch 2 + fa 2 + gb 2 + ch 2 = 0 ; 
that is, the moment of each pair of lines is = 0. It may be remarked that the ratios 
X : X 1 : X 2 may be determined from any two of the six equations 
Xu -f- XjUj + X 2 u 2 — 0, ... Xh -f- Xj/q -1- X 2 h 2 = 0 ; 
but that in consequence of the moments being each = 0, there is not for the deter 
mination of these ratios any such set of equations as occur in the cases subsequently 
considered of a threefold relation, &c. 
39. In what follows we have three or more sets (cq, b 1} c 1} f, g 1} /q), &c.; and we 
may without loss of generality assume that each of these are the coordinates of a line : for 
replacing the several coefficients cq, ... by linear functions /qtq + g 2 a 2 + g 3 a 3 + «fee., &c., the 
multipliers may be determined so that these are the coordinates of a point: and since 
for each set there is only a single condition to be satisfied by the two or more 
ratios /q : g 2 : g 3 ... , it is easy to see that no cases of failure will arise. 
Case of tlie threefold relation. 
40. The threefold relation is 
u, 
b, 
c , 
f> 
9 » 
h 
= 0, 
ttl, 
b 1} 
c i> 
fi> 
9i> 
hi 
u 2 , 
h, 
c 2 , 
u 
9-2, 
K 
a 3 , 
b 3 , 
/3, 
9s, 
h 3 
where (cq,...), (a 2 , ...)(u 3 ,...) are each the coordinates of a line. Here writing 
Xu + XjUi -f- X 2 u 2 + X3U3 = 0... , 
it is clear that every line which meets each of the lines (uj,...), (u 2 , ...), (u 3 , ...) will 
also meet the line (u, b, c, f g, h)\ the lines which meet the first-mentioned three lines 
are the generating lines of a hyperboloid having these three lines for directrices, and 
it hence appears that the line (u, b, c, f g, h) is any directrix line whatever of the 
hyperboloid in question. 
41. Using the notations 01, 02, 12, &c. to denote the moments of the several pairs 
of lines, viz. 
01 = af + b g 1 + c Jh + f Uj + g + h c u 
12 = d\fi~f b x g 2 + C\h 2 J rfa 2 + g x b 2 + /q c 2 , 
&c., 
then from the equations Xa + \ 1 a 1 + \ 2 a 2 +X 3 u 3 = 0, &c., we deduce 
X x 01 + X 2 02 + X 3 03 = 0, 
X10+ . + X 2 12 + X3I3 = 0, 
X 20 + Xj21 . + X 3 23 = 0, 
X30 + X x 31 + X 2 32 . =0,