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VIIL Quaternions and Projective Geometry. 
By Professor Charles Jasper Joly, Royal Astronomer of Ireland. 
Communicated by Sir Robert Ball, F.R.S. 
Eeeeived November 27,—Read December 11, 1902. 
Introduction. 
A. QUATERNION q adequately represents a point Q to which a determinate weight is 
attributed, and, conversely, when the point and its weight are given, the quaternion 
is defined without ambiguity. This is evident from the identity 
(A). 
in which is regarded as a weight placed at the extremity of the vector 
(B), 
drawn from any assumed origin o. It is sometimes convenient to employ capitals Q 
concurrently with italics cq to represent the same jioint, it being understood that 
(C). 
Thus Q represents the point Q affected with a unit weight. The point o may be 
called the scalar point, for we have 
(D). 
In order to develop the method, it becomes necessary to employ certain special 
symbols. With one exception these are found in Art. 365 of ‘Hamilton’s Elements 
of Quaternions,’ though in quite a different connection. We write 
(a, h) = 6Sa — aS5, [a, 6] = V. YaYb 
and in particular for points of unit weight, these become 
(a, b) = B — A, [a, b] = V.VaVb = V.Va . (b — a) . . . (F). 
Thus (ab) is the product of the weights into the vector connecting the points, 
and [a6] is the product of the weights into the moment of the vector connecting the 
points with respect to the scalar point. The two functions (a5) and [a6] completely 
define the line ab. 
VOL. COL—A 338. 
20.6.03 
