RFACE 
[570 
570] 
IN FIVE-DIMENSIONAL SPACE. 
81 
;hen the system of lines 
lines which pass through 
in the common plane of 
space the intersection of 
j planes at two harmonic 
we have on the quadric 
swering in ordinary space 
Derline of the other kind 
erve that, as regards the 
>etween the two kinds of 
two systems of generating 
Thus considering any two lines, the equations may be written 
u = ax + (3y, u = — {ax 4- by), 
v = ax + ¡3'y, v = — k {ax + b'y), 
where the lines will be of the same kind or of different kinds, according as k is 
= + 1 or = — 1. Observe that k is introduced into one equation only; if it had been 
introduced into both, there would be no change of kind. If the lines intersect we have 
{a + a)x + {/3 + b)y = 0, 
{a' + ka') x + {/3' + kb') y = 0, 
then answering thereto in 
it, and the lines through 
ine, that joining the two 
>oint. And, similarly, two 
taking two superlines of 
trough a given point, and 
in general any common 
however, the given point 
ine, but a singly infinite 
e and through the given 
of the two superlines, not 
viz. the condition of intersection is 
a + a, /3+6 =0, 
a + ka', (3' + kb' 
that is, 
a/3' — a'/3 + k {ab‘ — a'b) + a/3' — a'(3 + k {ah' — a'b) = 0, 
or, what is the same thing, 
1 + a/3' — a'/3 + k (1 + ab' — a'b) = 0. 
But we have, say 
a = cos 0, /3 = sin 0, a = cos <f>, b = sin <£, 
a' = — sin 0, /3' = cos 0, a' = — sin (f>, b' = cos <f>, 
and thence 
comparison, first the case 
lation of the surface may 
a/3' — a/3 = cos (# — </>) = ab' — a'b, 
and the equation is 
(1 + k) {1 + cos {0 — <£)} = 0, 
ations of a line on the 
viz. this is satisfied if k= — 1, i.e. if the lines are of opposite kinds, but not if & = + l. 
And it is important to remark that there is no exception corresponding to the other 
factor, viz. if k = + 1, and 1 + cos {0 — <f>) = 0, for we then have 0 — <£ = it, cos 0 = — cos 0, 
sin cf> = — sin 0, and consequently the two sets of equations for u, v become identical; 
that is, for lines of the same kind a line meets itself only. 
>n, viz. we have a 2 + /3 s = 1, 
lently a/3' — a'/3 = + 1 ; and 
e sign is + or —. It is 
vrite the equations 
Passing to the five-dimensional space, the equation of the quadric surface may be 
taken to be 
u 2 + v 2 + w 2 — x 2 — y 2 — z 2 = 0, 
and for a superline on the surface we have 
u = ax + (3y + yz , 
v = ax + /S'y + 7'z, 
w = a"x + /3"y + 7 "z, 
the other kind, according 
where (a, /3, 7), &c., are the coefficients of a rectangular transformation; the determinant 
formed with these coefficients is = +1, and the superline is of the one kind or the 
C. IX. 11