14 
Proceedings of the Royal Society of Edinburgh. [Sess. 
and 
X- X. 
V-Vq 
m 
t-t,' 
The homogeneous co-ordinates of the points in which these lines meet the 
hyperplane at infinity are evidently (I, m, n, q, 0) and (1', m, n', q, 0) : and if 
these points are conjugate with respect to the absolute whose equation is 
we must have 
ll' + mm + nn — qq' = 0 , 
which is therefore the condition that the two lines should be orthogonal. 
Now let a plane To through a point y^, t^) have the equations 
f — Xq) -K ~ 2 /o) + ~ ^o) “f" “ ^o) ” ^ 
If a line 
\ Tj^ix X.Q) + rji^{lJ ^o) ~ 
y - // q 
m 
t- L 
(H) 
I m n 
lies in this plane, we must have 
y-J' + ~ ^ 
Let a second plane tt^' through the same point {x^, y^, % t^) be given by 
the equations 
J “ ^o) ^2 ~ yo) ^3 (^ “ ^o) ^4 (^ ~ ^o) “ ^ 
I yi'{x - ^o) + y-iiy - yo) + - ^o) + - h) = 
If a line 
y-yp 
m' 
z-z, 
^ ^ ^ 
lies in this plane, we must have 
+ 4'^^' + is'n + iiQ' = t> 
:=:) 
, . n . • • • • (12) 
r)^ I + 7^2 m + 7/3 w +7]^q = 
If the planes ttj and 7n' are half-perpendicular, it must be possible to find 
values of I, m, n, q, satisfying equations (11), such that the equation 
IV + mm' -f- mi - qq = 0 . . . . .(13) 
is true for all values of l\ m\ n\ q, which satisfy equations (12): that is, 
equation ( 13 ) must be a linear combination of equations ( 12 ), and therefore 
we must have 
I = + fx'q{ 
m = X^2 + /^'*?2 ( 14 ) 
n = -F- /X773' 
q = - A^/ - fx-q^ 
where X and ju are as yet undetermined. 
