81 
435] 
ON THE SIX COORDINATES OF A LINE. 
and hence eliminating A, A 1} X 2 , A 3 , we find 
. 01, 02, 03 
= 0, 
10, . 12, 13 
20, 21, . 23 
30, 31, 32, . 
a relation between the moments satisfied in virtue of the given threefold relation; but 
which as a mere onefold relation is of course not equivalent to the threefold relation. 
It will subsequently appear that the equation expresses that any one of the four lines, 
say the line (a, b, c, f g, h) touches the hyperboloid having the other three lines for 
generatrices ; this condition is satisfied in virtue of the threefold relation which, as we 
have seen, expresses that the line (a, b, c, f g, h) lies wholly in the hyperboloid in 
question. 
42. The last mentioned determinant is the Norm of 
/01723 + /02.31 + /03.12; 
so that the equation may be written 
/01723 + /027 31 + /03.12 = 0, 
or, what is the same thing, 
/01/23 + /02 /31 + /03/12 = 0, 
it being of course understood that the signs of the radicals must be determined in 
accordance with this equation; we then find 
A : Xj : X 2 : A 3 = /23.31 . 12 : /02.03 . 23 : /03.01 . 31 : /01 . 02 . 12, 
or say 
= /23 /31 /12 : /02 /03 /23 : /03 /01 /31 : /01 /02 /12 ; 
in fact, substituting these last values in the linear equations for A, A x , Aa, X 3 , we find 
that the equations are all satisfied in virtue of the single equation 
/Ol /23 + /02 /31 + /03 /12 = 0. 
43. We have here 
Case of the twofold relation. 
ft , b, 
c, 
/> 
9 > h 
a 1 , b u 
c i> 
gu k 
b 2 , 
c 2 , 
u 
g 2 , ho 
«3, K 
C s , 
A> 
g 3 , k 
ft 4 , b 4 , 
c 4 , 
u 
g*, k 
= 0, 
where (a 1? ...) (a 2 ,...) (a 3 ,...) (a 4 ,...), are each the coordinates of a line. Here, writing 
Aft -p Ajfti T A 2 ft 2 + A 3 ft 3 4" h, 4 a 4 0 , 
C. VII. 
11