426 
Y, lying on the straight line X,X, determines a triplet, the other 
two members Y, and Y, of which are collinear with X,. If Y, 
describes the line X,X, the line y,= Y,Y, revolves about the fixed 
point X,. The triplets of the involution therefore determine polar 
triangles with respect to a conic’). 
7. The pencil (Zr) determines in the congruences [«°] and [@°] 
two surfaces A4 and Ag (§ 6) and conjugates to each curve a’ of 
A4 a curve #* of Ap. 
Now consider the cone generated by the chords which can be 
drawn from A, to the curves p° lying on Az. 
A plane @® through A, intersects Ag along a curve c° which 
contains oo! triplets of the involution (X)*. If the point X, describes 
the curve c° the line X,X, envelops a curve of the fifth class, every 
tangent of which is a chord to one of the curves g° lying on Ag. 
Hence through A, pass five of these chords, whence it follows that 
the just mentioned cone of chords through A, is of degree five; it 
will be denoted by K°. 
Now consider a ray tf, of (7\z) as a chord of an a’. The quadrie 
cone which projects this curve from A, has ten edges a, in common 
with #&*. Each of these rays a, is a chord to a 8° which has also 
one of the rays, fs, of the pencil (¢) among its chords; to each ray 
le therefore are conjugated ten rays fz. 
A ray ts is a chord to a definite curve 8°; this curve sends a 
chord a, through A,. The curves @* which have a, among their 
chords, form a cone a,°; this cone intersects t along a conic on 
which the points of transit of the curves «° determine an involution 
[*. The triplets of chords joining these transits two and two envelop 
a second conic. Hence two of these chords pass through 7. To a 
ray tz therefore correspond two rays tz. 
(Tr) evidently contains twelve rays ¢,= ts. To each of these the 
involution /*° conjugates a ray a,; on the ruled surface (¢’) corres- 
ponding to the pencil (4) A, is therefore a twelve-fold point. Hence 
the points of intersection of the ruled surface (¢’) and the line a,, 
lie on. 12 lines 4,42 lines a, lines ofc rss of lei NO INGEN 
and on 6 lines which do not lie in any of these planes and do not 
pass through A, or 4. 
A plane pencil of lines is by the transformation (tt) converted 
into a ruled surface of degree 39. 
This ruled surface has 10 twelve-fold points (Ar, By). 
The surfaces A4 and Ag each have a curve of the fourth degree 
!) This conic is the locus of the points whereat & is touched by the curves «5. 
