289 
PEOFESSOK C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
98. We shall now explain a method which promises to be of considerable value in 
the application of quaternions to projective geometry, 
A. bi-linear quaternion function J (pq) is a tunction of tAvo quaternions (p and (j) 
linear and distributive Avith respect to both. It may be reduced to the form 
/(PT) = . . . . (409), 
Avhere cq, cq,, cq, cq are any four quaternions and Avhere /j, /Ij, andare four linear 
quaternion functions. The bilinear function involves sixty-four constants, sixteen for 
each of the fmr functions. 
99. Writing generally for all quaternions and q 
f{pl)=fA<lP) .(110), 
we may call the iieAv bilinear function /) the qjermulate of the function /. When a 
function is unaltered liy transposition of the (piaternions, it may be called a pvr- 
mulable function. Thus 
iP'l) = Lf{P<l) + y] (pq) . (Ill) 
is a permutable function, the pernmtable part of/'or/). A permutable function 
involves forty constants, the functions of (409) being then self¬ 
conjugate, 
100. When a bilinear function changes sign Avith transposition of its quaternions, 
it may be called a comhinatorial function. Thus 
^ {P<j) = - i// {P<l) .(412) 
is combinatorial. It vanishes for = <j, and, regarded geometrically, it relates not to 
a pair of points, but to tbe line joining the points. 
A bilinear function is thns reducible to tlie form 
J (P7) = (at) + ^ {pu) ; ,/) ( at) = P ( at) ~ ^ (at) • • (ns); 
and is uniquely resoluble into its permutable and combinatorial parts. 
101. Writing generally for any three ([uaternlons, p, q, and /•, 
Sf/’( at) = ^pf y) = yy {pa .(m), 
we shall call the neAA' functions (yiq), (^>q) the first and second conjugates 
j{pq). In i-dci jfipq) is the conjugate Avhen the first quaternion qi alone varies, and 
J" iP'l) 11^6 conjugate Avhen the second Auiries. 
102. As the accents employed to denote the permutate and the first and second 
conjugates are not commutative in order of applicatimi, it is safer to use brackets in 
the rare cases in Avhich double accents are necessary. Thus 
/(at) = (./7(at) = (./'T(at) = (.4 (at) . ■ • . (415), 
2 p 
VOL. cor. 
A. 
