280 PEOFESSOE C. J. JOEY OX QUATEEXIOXS AXD PEOJECTIVE GEOMETEY. 
systems. If a is the pole of the plane = 0 with resj^ect to one of the quadrics, 
a and h are connected hy the equation 
I = a, or (/+ t)h = (/+ .s)o, 
/+ t 7 
or a = ~ 0 
J+s 
(341). 
Given b, the locus of a is a twisted cubic if alone varies, a right line if s is con¬ 
stant, and a quadric 
[afa Ifb) = 0 .(342) 
when s and t are both variable. (Compare Art. 70.) 
The points of contact of the plane with quadrics of the system are found by adding 
the condition = 0, when we find three points, one point or a conic locus. 
A generalized normal joins a point to the reciprocal of its tangent plane, thus for 
u variable, 
<1 
J + ^ 
when 
Sa4+-a = 0 
y +1 
(343) 
is the genertilized normal at the j^oint a ; or deleting the condition and allowing 
t and u to vary, we have the equation of the assemblage of normals through the point 
a, and wlien a itself varies, we see that (342) represents the complex of normals to 
the system. 
82. In general, two quadrics h G hitersect in a curve through wliich no 
third quadric of the system can pass, but wlien q = 4, an infinite number of the 
quadrics intersect in the curve. This follows from the consideration that 
Bq . 
^]){ f fi- y{ f -{- ■yj)_(/ + q) _ 
(/ + h) (/ + ^ 2 ) 
^7 = 0 
(344) 
is the general equation of a quadric through the curve ; and a factor will not cancel 
unless q = q. 
If q is any point on the curve of intersection, the poles of the tangent jjlanes at 
that point with respect to some third quadric of the system will be conjugate to that 
quadric if 
Bq 
(/+ ■ 4 ) if + --^a ) ( / + q) _ A 
(/ + h) (./ + G) (/+ %) ^ 
(345). 
In order that this may be the case for every point on the curve, the factor f ^5 
must cancel. Ihus wm must have s.^ equal s^, s.i or q. But further, on comparison 
with ( 344 ), it appears that the third quadric must coincide with one of the others, or 
else that q = q and = .s,,. 
I his theory embraces the law^s of confocals, their orthogonal section, and the pro¬ 
perty that the pole of the tangent plane to one, at a point of intersection with a 
second, taken with respect to the second, lies in its tangent plane at the point. 
83. More generally, given any three quadrics 
