250 PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
When c coincides with a point a on the line, let d be the jDoint for which (Art. 15) 
(F^ — QFo)« = F.a .(148), 
then SaF^a = 0 , and S 6 Fo« = 0 , and the line ab touches the two quadrics at a. The 
conics in the common plane sections touch (type II,,). 
If, further, d coincides with the point a (type IT^), the point Faff is derived by the 
ojieration of Fj — QF.j from some other point a" (Art. 32), and therefore 
SaFoA = 0 ; S 6 Focf = 0 ; and SaF^a' = 0 ; S 6 Fpf = 0 . . (149). 
Hence it appears that the line d + xh meets the two quadrics in the same two points, 
and the lines from a to these points are common generators. The intersection of the 
quadrics consists, therefore, of a pair of lines and a conic passing through their 
common point (type II^). 
Finally, it remains to notice the case of a plane locus of united points with the 
fourth point in the plane (HL). It may be proved that in this case the coincident 
plane sections consist of a pair of lines along which the quadrics touch. 
35. Summing ujj, tlie intersection of two quadrics according to the types of the 
function F^^T^, is 
I^, a twisted quartic with two appa,rent double jDoints ; 
I 3 , a twisted quartic with three apparent double points ; 
I 3 , a twisted quartic with two apparent and one real double point; 
Ij., a right line and a cubic touching it ; 
I 5 , a right line and a cid^ic ; 
II^, two conics; 
IL, a pair of lines and a conic ; 
Ilsf two conics in contact; 
II^, a pair of lines and a conic through their intersection ; 
IIIj, the surfaces touch along a conic ; 
IIL, the surfaces touch along two generators ; 
lA^, the intersection is a quadrilateral. 
SECTION V. 
The Square Root of a Linear Quaternion Function. 
Ai’t. Page 
36. The sixteen square roots of the general function.250 
37. Case of a function with loci of united points. 251 
38. Various useful formulse.252 
39 A square root of the conjugate is the conjugate of a square root.252 
36. When the same eftect is produced b}" the twice-repeated operation of one linear 
quaternion function and by the single operation of another, the foiauer may be said 
to be a square root of the latter. 
