PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 227 
3 . In order to specify a function of this kind it is necessary to know the quater¬ 
nions a', V, c, d' into which any set of four unconnected quaternions, a, h, c, d, are 
converted. Thus, from the identical relation 
q{ahcd)a{bcdq)h{cdqa)c{d( 2 ah)d{qabc) = 0 . . . (5), 
connecting one arbitrary quaternion with the four given quaternions, is deduced the 
equation 
fq{c(bcd) + a'{bcdq) + b'{cdqa) -j- c'{dqab) + d'{qabc) = 0 . . . (G), 
which determines the result of operating by f on q. 
When we are merely concerned with the geometrical transformation of points, the 
absolute magnitudes^' of the representative quaternions cease to be of importance, 
and the function 
fq = a:A'(BCDq) + yB'(cDqA) + 2c'(DqAB) + !(W(qABc).... (7), 
which involves four arbitrary scalars, converts the four points A, b, C, d into four 
others, a', b', c', 'd. Given a fifth point e and its correspondent e', the four scalars 
are determinate to a common factor, and subject to a scalar multiplier, the function 
which produces the transformation is 
/■ // \ (bWdV) , // \ (Gd'eW) I // \ (dTWbT 
fq = A'(BCD5) . + B (cDJA) . ' + C (d.JAB) . 
(e'aVc') 
+ ^ ■ (ilio. 
• ( 8 )- 
It is only necessary to replace q Ijy e in order to verify this result. 
4. A linear quaternion function, f, being regarded as effecting a transformation oj 
points, the inverse of its conjugate f'~^ produces the correspionding tangential trans¬ 
formation. 
For any two quaternions, 'p and q, 
^pq = ^p)f-^f = S/'-ip/ = ^p(f if q' = fp p^— f'-^p . . . (9). 
Hence any plane '^pq — 0, in which the quaternion cq represents the current point, 
transforms into the plane ^p/[' = 0, and the proposition is proved. 
Thus, when symbols of points (q) are transformed by the operation of j, syml^ols of 
planes ( p), or of points reciprocal to the planes, are transformed by the operation 
5. Hamilton’s Ijeautiful method of inversion of a linear quaternion function 
receives a geometrical interpretation from the results of the last article. 
* Ill accordance with the notation proposed (‘ Trans. Roy. Irish Acad.,’ vol. 32, p. 2), capital letters 
are used in this article concurrently with small letters to denote the same jioints, but the weights for the 
capital symbols are unity; thus q = qS(/ = (1 + OQ ) Hq. 
2 G 2 
