248 PEOFESSOR C. J. JOEY ON QUATERNIONS AND PR0JECTI\T: GEOMETRY. 
Let c and d be the remaining united points. By (135) the line c, d lies in the 
common tangent plane; so in order to determine the generators of the two quadrics 
in the plane, it is only necessary to determine the points in which the quadrics meet 
the line c, d. For the first and second quadrics, the equations determining tlie 
points c + xd are respectively (134) 
tgScFoC + .r'QSdFod = 0 ScF^c + x^Sc/Foc/= 0 ; . . . (137). 
The quadrics consequently have distinct generators unless Q = Q, and unless the 
points c and d are distinct. 
For quadiics having a pair of common co-planar generators, F^'T^ is of the type 
TIo, and conversely. 
32. In the next place, let tliree roots q be equal, so that a is the union of three 
united points of /= Fo~^Fj. The point a of Art. 15 (78) is now in the common 
tangent plane, because it bas been derived by the operation of f — q from another 
point In fact we have 
(Fi-qF 2 )«" = Foa, (Fg-qFo)rt'= F. 3 n .... (138); 
and from the first of these it is obvious that S«Fo«' = 0 (= q“^S«F^a'), while the 
second may be written in the form 
Fi (a + xd) = (q + x) 
xt^ 
h + 
a 
X 
(139). 
This equation shows that the polar plane of the point a -j- xd with respect to the 
* * * * 
first quadric is identical with the polar plane of a-\ -^ with respect to the 
q fi- aj 
second; and because d lies in the tangent plane, in the limit where x becomes 
infinitesimally small, tlie two points l)ecome identical to the first order of x, and the 
common polar plane becomes a consecutive tangent plane to both qiiadrics. The 
(piadrics have, therefore, stationary contact, and their function Fo“^Fg is of the 
class I3. 
The generators in tlie?^tangent plane are now found by expressing that xd + d is 
on one of the quadrics; the equations may be written in the form 
.x~qSrt FoA + 2.rqSAF3d + qSdF.d = 0 ; x~^dF,d + 2.rSr7Foc/ + SdFod = 0 . (140), 
where the equation for the first quadric has been reduced by the aid of (138), in 
order that it may be compared with that for the second quadric. The generators are 
common if, and only if, q = q, and the function is tlien of the type II^^. 
33. When the four united points coincide, the point a" as well as d lies in the 
common tangent plane, a" having been derived, as d was in the last article, from a 
third point a". From the three equations 
