1159 
If the pencil is represented by 
7+, Ha, + d4¢7 = 0!) 
the tangential plane in (y) has the equation 
ye, + yet, + y,e, + Aye, = 9, 
4 being determined by 
Sie zi Va Te HAI, = 9 DIK sein Mer tal toate it (1) 
So for its coordinates (1) we find 
NY = Nat Ya = N32 Vg == Nat AY,» ae acne 
or 
My? Ys = Ne? Yas = Nn Aas = Mt YY? + Wars ys)» -1 (8) 
From (1) and (2) we deduce 
: ús 
ntt + 45" + =0, 
HMM = Yo? Nas = Yo? MM, SY: — (M+ 2” + 757) - (4) 
So (4) shows that any plane has only one null-point. 
If the null-plane (#) passes through the fixed point (er) we 
have > zene == 0, so the equation of (P,’ is 
Be HWI T Aaa SZH + Ys HH) 0 >» - 5) 
The intersection of this surface with the surface belonging in the 
same way to the point Q(w x) breaks up into the singular conic 
y, = 0, GRT In 0 
and a second conic lying in the plane 
(zw, — zw) ¥, + (2,0, — 2,0.) Ya + (2,6, — 240) y, = 0. 
The latter contains the null-points of the planes passing through 
PQ. From this ensues that y is equal to one. ”) 
All the surfaces (P)* pass through the principal point y,=y,=y,—0. 
As could be expected, this point is the vertex of the quadratic cone 
touching all the surfaces of the pencil (®*) along o’. 
The null-planes (7) of the points (y) of the plane & envelope the 
quadratic surface 
Gin, + Eman + Eens, = 54 (M7 + 00° + 19°). 
All these surfaces forming a system co? touching the plane 1, = 7, = 
7, = 0 (w,= 0) and the quadratic cone with the equation 4, = 0, 
1) Coefficients which might present themselves have been comprised into the 
definition of the coordinates. 
*). This can also be found by considering the involution determined on PQ by 
the pencil (7); one of the coincidencies lies in the plane of the conic ;2, the 
other is point of contact with one of the quadratic surfaces. 
