324 PROFESSOR C. J. JOTA' OX QUATERXIOXS AXD PROJECTIVE GEOMETRY. 
SECTION XXL 
The Extensiox of the Metiioi) to Hyper-Space. 
Art. 
159. The equation of a flat in terras of parameters. .304 
160. The combinatorial equation of a flat... 304 
161. The reciprocal of a flat.305 
162. The symbol of a flat.. 
163. The symbol of the reciprocal flat. 326 
159. Exactly as in quaternions we may regard the sum of a scalar and a line 
vector in space of n dimensions as the symbol of a weighted point. 
If 
g = S^ + = !^l 
+ ~ '^5' i 
(57G); 
q is the symbol of the point Q to which a weight S(/ is attributed. 
The point represented by a sum of point symbols is the centre of mass of the 
^velghted points, and the veeight attributable to that jDoint is the sum of the weights. 
The equation 
q = a-\-th .(577), 
in which t is a variable scalar, is the equation of the line ah. 
I he most general homographic divisions on two lines ah and cd are repre¬ 
sented by 
q = a-\-th, q~ c-^ td .(57«), 
in which the weights Sn, 85, Sc, M have been suitably selected. 
The equation 
q = t.^a.^ . ..( 579 ) 
represents the plane of tlie points cq, cq, cq; and more generally 
q = qcq + qrq + &c. . . . + CYC«.(580) 
is the equation of the (m — l)-flat containing the m points cq, a.-, . . . a,„. 
I believe it is more convenient to call generally a plane space of m dimensions an 
m-flat, and to retain the ‘plane for its ordinaiy signification—a two-flat. 
100. In accordance with Hamilton’s notation (‘Elements,’ Art. 365) Ave propose 
to Avrite 
[n/qcq . . . o J = \p^.L'cqL cq . . . V(h„-“Xfl;;V,„_^.V(qVtq . . . Vn,„Scq 
or briefly 
[o]vi = \ „ [a]„ -b ^ [n],„. 
as the sjunbol of the (?n l)-flat containing the m points rq, n.-, . . . a„,. 
(581) ; 
(582) ; 
