PEOFESSOE C. J. JOEY ON QUATEENIONS AND PEOJECTIVE GEOMETEY. 249 
cr = - ^^F.) a" = F,a'; (F^ - ^.F^) a = F.a . (141), 
we see that, in addition to the conditions that the points should lie in the tangent 
plane, we have 
Sa" (F^ — fjFo)^ = 0 ; Sa'FgO.' = 0, and Sa'F^a =0 . . (142), 
as appears from operating on the third by Sa" and using this result in operating on 
the second by Sa', and finally operating on the third by Set'. The line a + xa is 
consequently a generator of both quadrics, and the function belongs to the class 
The remaining generators, determined by the point in which a + ya" meets the 
surfaces again, are deducible from the equations 
qSa Fy:F + yt,Sd'F,a" + ySd'F.d = 0 ; SefF^a" + ySd'F.a" = 0 . (143). 
If these remaining generators are common to both quadrics we must have 
Scf'Focf = 0, and then they coincide of necessity with the other generator, and the 
quadrics become a pair of cones touching along a generator. 
34. Suppose the function to have a line locus of united points, so that 
Fi« = t^F^a ; F^h = t^F^h .(144); 
it immediately follows that one quadric meets the line a, b in two points common to 
the other, and the quadrics touch at these two points. Suljstituting in the equations 
of the quadrics 
q z= xa 4- yb + 2 (c + tal) .(145), 
the equations become, 
qS (xa + yb) Fo {xa + yb) + z' (qScFoC + ?(,^qSc/Fnd) = 0 
S (a;« + ^6) Fg (x?^ 4 y6) + 2^ (ScFgC 4 trSc/Fy/) = 0 .... (146), 
and for a constant value of ii, these represent the sections by an arbitrary plane 
through the line a, b. These sections are identical if 
(q — q) ScFgC 4 iF (q - q) St/F// =0.(147), 
and as this is a quadratic in u, the quadrics have two plane sections common. The 
function f is of the type II. The case of coincidence of the points c, d has occurred 
in Art. 31, one of the conics breaking up (type IF). 
If q — q, while c is not situated on ab, the quadrics have two coincident jDlane 
sections, or ring-contact. The type of the function is IIIj. 
If q = but c not coincident with d, the function is of the class lY., and the 
quadrics intersect in common points on the line c, d. Let ab meet the quadrics in 
4, b' and cd in c'd', then it is very easy to see that 4, c', V, d' is a quadrilateral 
common to both surfaces. 
VOL. CCI.—A. 
2 K 
