226 PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
SECTION I. 
Fundamental Geometrical Properties of a Linear Quaternion Function. 
Art. 
1. Definition of a linear quaternion function. 
2. The general linear transformation effected by a linear function. 
3. Specification of a function by four quaternions or five points and their deriveds. 
4. The transformation of planes effected by the inverse of the conjugate function 
5 . Geometrical interpretation of Hamilton’s method of inversion 
6. Geometrical illustration of the relations connecting Hamilton’s auxiliary functions 
7. Ihe united points of a linear transformation. 
8. Relations connecting the united points of a function / with those of its conjugate /' 
9. Introduction of the functions /o = I (/+/'), f, = h{J-f') . 
10. Sf//oy = 0 and fiqf,p = Q represent the general quadric surface and the general linear 
complex. 
11. The pole of a iJane to the quadric is/o-^G and the point of concourse of lines 
of the complex in the plane . 
12. The united points of/^ form a quadrilateral on the sphere of recijirocation .... 
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I. Tlie quaternion equation 
f{p + 9)=fp+fq .( 1 ), 
may be regarded as a definition of the nature of a linear quaternion function f, the 
quaternions p and q lieing perfectly arbitrary. As a corollary, if x is any scalar, 
f{xp) =xfp .(2), 
and on resolving fq in terms of any four arbitrary quaternions a^, a. 2 , %, a^,, we 
must have an expression of the form 
fq = af>h^q + af>h 2 q + af)h^q + af>hpi .(3), 
because tlie coefficients of the four quaternions a must be scalar and distriliutive 
functions of q. Sixteen constants enter into the comjiosition of the function f, lieing 
four for each of the quaternions h. 
2 . When a quaternion is regarded as the symbol of a point, the operation of the 
function y’jiroduces a linear transformation of the most general kind. 
Tlie equations 
f{xa -b yh) = xfa + yfh ; f{xa -f- yh + zc) — xfa + yfh + zfc . . (4), 
show that the right line n, h is converted into the right line fa, fb, and the plane 
containing three points a, h, c into the containing their correspondents, fa,fh 
and fc. 
The homographic character of the transformation is also clearly exhibited. 
