PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 325 
In order to justify this projDOsal, we oliserve that the array [<x^(Xo . . . changes 
sign whenever two contiguous elements are transposed. It consecpiently vanishes 
whenever one element is a scalar multiple of another, or whenever any group of 
elements is linearly connected hy scalar coefficients 5 and it does not vanish under 
any other conditions. It is ecjuivalent to the most general one-row array that can 
be formed from the w symbols o, because, according to the principles laid down on 
the subject of quaternion arrays, the general one-row array must be of the form 
{opQ. . . a4 = xY„,.Ya,Ya ,. . . Vo, + ijS ± Y,„_,.Ya,Ya ,. . . Vo,,Sai . (583); 
and the separable parts and of \ci\ni- afford all the information contained in 
the general array with indeterminate scalars x and y. 
The equation of the flat containing m points a may be written in the form 
. o J = 0 .(584), 
as this implies (580) 
q — Qcq -{- tyio . ff- 
in which Q, Q. . . t„i are variable scalars. 
161. Returning to the relation (582) 
\jY\n — Y„i \ct\ni “b 
it IS evident that is equal to the product of a scalar and a set of m — 1 
mutually rectangular unit vectors is - - - im, iu the (m — l)-flat containing the m 
points a^, 0.2 - . . a„,. It is also apparent that Y„, [«],„ is the product of a set of m 
mutually rectangular unit vectors in the w-flat containing the origin and the jjoints 
a multiplied by a scalar. We may take this product of m vector units to be 
iy^s • • • im- Thus we have 
[«]« = (Vh — --‘im .(585), 
where t.Q + qf, = 0, — 1, and where x and y are certain scalars. (Comjiare 
Cliffoed’s ‘Mathematical Papers,’ p. 398.) 
From this we find the symbol of a definite point 
A„ 
— 1 I \/Y\m _ 1 
Xl.ls - - ■ 
yhUis - - - I 
rr __ X - 
• — Id— h 
Vh y 
. (586); 
and we verify at once that this point is a conjugate of all the points a with respect 
to the quadric 
= 0. (587), 
because for any one of these points we have 
S.,A,„ = Sc. (1 + 4 .EI) = Sc. + ^ 4 - 4 - = Sc. - Sc. = 0 . (588), 
as appears on reference to the equation (581). 
