PROFESSOR C. J. JOEY OY QUATERNIONS AND PROJECTIVE GEOMETRY. 225 
replacing the quaternions by 1 + «, I + 1 + 7 , and 1 + S respectively. In 
complicated relations it may be safer to separate the c|uaternions as in these formulae 
by semi-colons, but generally the commas or semi-colons may be omitted without 
causing any ambiguity. 
These new interpretations are not in the least inconsistent with any principle of 
the calculus of quaternions. We are still at liberty to regard a quaternion as the 
separable sum of a vector and a scalar, or as the ratio or product of two vectors, or 
as an operator, as well as a symbol of a point or of a plane. 
In particular, in addition to Hamilton’s definition of a vector as a right line of 
o-iven direction and of given magnitude, and in addition to his subsequent interpre- 
tations of a vector as the ratio or product of two mutually rectangular vectors, or as 
a versor, we may now consider a vector as denoting the point at infinity in its 
direction, or the plane through the centre of reciprocation. For the vector OQ of 
equation (B) becomes infinitely long if Sq = 0 , and the plane Slq = 0 passes through 
the scalar point if S? = 0 . We may also observe that the difference of two unit 
points A — B is the vector from one point b to the other a, and this again is in 
agreement with the opening sections of the “ Lectures.” 
Additional illustrations and examples may he found in a paper on “ The Interpre¬ 
tation of a Quaternion as a Point-symbol,” ‘Trans. Pvoy. Irish Acad.,’ vol. 32, 
pp. 1-16. 
The only other symbols peculiar to this method are the symbols for quaternion 
arrays. The five functions (ah), [ah'], (abc), and {ahcd) are particular cases of 
arrays, being, in fact, arrays of one row. In general the array of m rows and n 
a, a, «3 . 
a,-. 
6^ 63 63 
( 0 ) 
I Pi Ih P3 • • • P 
U J 
may be defined as a function of mn quaternion constituents, which vanishes if, and 
only if, the groups of the constituents composing the rows were connected by linear 
relations with the same set of scalar multipliers. In other words, the array vanishes 
if gcalars q, Q . . . t„ can he found to satisfy the m equations 
"b d“ • • • "h = 0, 
F ■ • • tiF; = o. 
hPi “h QPs F • • • "h 
The expansion of arrays is considered in a paper on 
Boy. Irish Acad.,’ vol. 32, pp. 17-30. 
VOL. ccr.—A. 2 G 
“ Quaternion Arrays,” ‘ Trans. 
