niOFE^^SOn C. J. .TOLY OX QUATELXTOXS AND rrKATECTIYE OEOMETRY. 293 
Art. Pac'e 
, O 
118. The tliree tangent planes at tliis point.297 
119. Analogies with a Steinkr’s quartic.298 
120. The lines in the y space which transform into plane cnhies in the p space.299 
121 . Ihnes in the p space transform into twisted sextics which touch the critical curve in 
ten points.299 
122 . The locus of dacobian correspondents of points on the critical curve is of the tenth 
order, and every Jacobian quartic contains ten lines..‘300 
12.9. The relations coimecting the orders of a surface, the complementary surface, and the 
surface in the p space into which lioth transform..300 
121-. The coiTcsponding orders for curves.301 
12.7. The curve of intersection of two scxtie sui faces.302 
126. Connectors of pairs of united iioints form a complex of the foui-th order.302 
127. The locus of the united points of functions having a zero root.302 
128. The loci of united points corresponding to double and ti'iple latent roots .... 303 
129. The united points corresponding to quadruple latent roots, and their nunibei' . . . 303 
130. The locus of united points of functions whose latent quarties are perfect s(|uarcs . . 301 
107. Wlienone of tlie (Riateriiioiis in a bilinear function is regarded as a ((uaternion 
parameter, the function represents a triply-infinite system of linear rpiaternion 
functions, or four-system of linear functions, to borrow a convenient phrase from 
Sir Robert Ball’s ‘ Theory of Screws.’ 
Thus 
f{pq) z= xj{iqq) + X.Jfpyj) + xjfpyi) + xjfpyi), 
wdmre p = x^p^xppx.pp-\-xpp^, . . (431) 
is a linear comlnuatiou of four given linear functions /{pj/), the rpiaternlons />„ being 
supposed given while the scalars x„ are variable. 
It is frerpiently of advantage to use the notation 
./■(y^v) =fM = fAp) .(432), 
when the bilinear function is regarded as a function of q or as a function of p. 
108. An arlnfrary point is a united point of a definite fund ion of tin' four-system, 
provided it does not he on a criticed curve o f the tenth order. 
If <1 is assumed to be a united point of a function determined by p, 
{.f{pAdl\-h, or f{pq) = t,/, or f{p) = l,/ . . . (433); 
and the solution of the equation in its tliird foiTu is 
2>J{q) = tFfq), or 2 > = Ffq), t = J(q) .... (434), 
where is Hamilton’s auxiliary function corresponding to and where J{</) is 
the fourth invariant of f (Art. (103)). 
Hiis solution is definite (Art. 15), provided q does not lie upon the critical curve 
F^ (q) = 0 
(435). 
