PROFESSOR C. J. JOLY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 247 
SECTION IV. 
The Relations of a Pair of Qltadrtcs, Sr/F^r/ which depend 
ON THE Nature of the Fltnction F^^'F^ 
Art. 
30. The self-conjugate tetrahedron. 
31. Coincidence of two of its vertices. 
32. Coincidence of three vertices. 
33. The tetrahedron becomes a point. 
34. The function r 2 “^Fi has a locus of united points. 
35. Scheme of the nature of intersection of the quadrics according to the type of the 
function F 2 ~^Fi... 
Page 
247 
247 
248 
248 
249 
250 
30. We shall briefly consider the relations of a pair of quadrics which depend on 
the peculiarities of the function F^''^^, where 
SgF^p = 0, SgFoq = 0.(133) 
are the equations of the two quadrics. 
If the polar plane of the point a is the same with respect to the two quadrics, 
F^n = .(134), 
where h is a scalar, because (Art. 11) the symbols of the polar planes are Fpt and 
FgU. Here is a latent root of the function Fo”Wj and a is a united point. 
If?) is a second united point answering to the latent root A, we have, on account of 
the self-conjugate character of the functions Fi and Fo, 
qS/jFort = S/)Fp( = SuF/) = t.SaYob = 0.(135), 
provided tlie latent roots are distinct. Thus the polar plane of n contains the points 
h, a, and d ; and the tetrahedron is self-conjugate to both quadrics. The function 
Fi“^F 3 belongs to the general type I^, in which all the united jioints are distinct 
(Art. 13). 
31. Let two united points a and h approach coincidence. The relation (135) 
remains true up to the limit, and ultimately 
ScH\a = 0, SaF 3 a = 0.. (136); 
and the coalesced point is situated on the curve of intersection of the surface. By 
(134) the symbols of the tangent planes to the two surfaces are identical, and the 
two surfaces touch. 
If then F 3 “W^ belongs to the type L, the surfaces touch, and conversely; and if 
the quadrics touch in two distinct points the type of the function is I 5 , and the 
intersection is a line and a cubic. 
