308 PEOFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
of the parameters, the ecjuations of a pair of lines transforming into conics in a 
common plane must be 
“1" '^ 0^3 “t" W 3 Q — 0, 
+ h + h = 0 . 
Wi 
• (512); 
this is a consequence of the harmonic section. Two lines thus related may be said to 
be conjugate, and there exist four self-conjugate lines 
h db dz i's — .(513), 
any one of which transforms into a conic having ring-contact with the quartic. The 
planes of these four conics transform back into cones, touching the plane along the 
self-conjugate lines. The self-conjugate lines join triads of non-corresponding 
Jacobian points, such as q,, -|- q^^ q.^ — q^. 
It is easy to see that the four conics are inscribed to the faces of a tetrahedron, 
and that each touches the other three. Consider, for example, the conics transformed 
from the sides of the triangle, q.. q^, q.^ -j- q^ -b q^. The equation of one conic is 
P =fiP2 + (Z 3 + ^ {<h + 'Zi), q.2 + 'Z.s + i (Ys + "Zi)) 
— 2 {Po + P-:s) + 2^ {pq +zq3 +Zbi + 1 ^ 13 ) + 2^' {po d"i^3i) • • (51-1) 5 
and this shows that the conic passes through a limiting point on eacli of two of the 
double lines; and as the pole of the chord is symmetrical with respect to the suffixes, 
it is likewise the pole of corresponding chords for the conics into which the other 
sides of the triangle transform. 
It is not difficult to prove that every line in the plane through one of the six 
Jacobian points transforms into a conic having a fixed tangent. The tano-ent for 
00 O 
the point q^ -f q^ is 
— ik d‘ih3 + ^ (Zbs+ibi).(515). 
138. Let a connector meet the Jacobian in the jioints a, a', h and c, a and a' being 
correspondents so that / (aa') = 0 ; let ?/ and c' be the correspondents of h and c; and 
consider the jioints of an octad in the plane The two connectors aa' and hh' 
in this plane intersect in the point h, and as h is its own harmonic conjugate with 
respect to 6 and two sides of the triangle of Art. 134 unite in the line aa'. Let 
be the harmonic conjugate of b with respect to a and a', then is a vertex of the 
infinitely slender triangle, the remaining two being the point h counted twice. 
(Compare Arts. 132 and 134.) 
The point being the intersection of tiie connector aa/ with a consecutive 
connector, is a focal p)oint on the ray aa' of the congruency (49G) ; and similarly 
the harmonic conjugate of c to a and a', is the second focal point ; and hy 
Hamiltox s theory the ray touches the focal surface at these two points.* 
* This theorem of the construction of the focal j^ioints is an extension of Mr. Russell’s theorem for the 
congruency of lines joining corresponding points on the Hessian of a cubic siu'face. R. Russell, 
“Geometry of Surface.s derived from Cubics,” ‘ Proc. Roy. Irish Acad.,’ vol. 5, p. 461. 
