( 569 ) 
are in four Ways projective to the eight singular rays (u); conse- 
quently through QO, and Q, pass four conies bearing each four 
points of intersection of two tangents out of ©, and O, and at the 
same ume four points of intersection of rays out of O, and O, to 
the points of contact of those tangents (double-rays of the (2, 2)). 
If O,O, is a branchray for both pencils, one of the four conics 
degenerates, in which case C, has cusps in OU, and O, (see my 
paper “On bicuspidal curves of order four’, Proceedings of the 
meeting of Dec. 24th 1908, Vol. IX, p. 499). 
We suppose that 0,0, is conjugate as double-ray to the branch- 
rays O,O, and O,O,. The equation of correspondence must then 
furnish for 20 and for «= 0 the equations u? =o and2?—=o: 
Heneeaaes dr Oa =O a = 0. 
The equation of C, can now be written in the form 
Bt, + 20,0,a, (0,2, + 5,2, + bewo) + 2,4 = 0. 
In each of the two double points one of the branches has an 
inflectional point; the corresponding tangents are wv, — 0 and x, = 0. 
Out of each of the two jlecnodal points three more tangents can 
be drawn to C,. They are represented by 
bite,’ + 26,b,4,?a, + (6,7—1) w,a,? — 2b, 2,' 
beta,’ + 26,5, w, + (b,7—1) a,a,? — 2b,2,' 
By eliminating z,* we find 
beta? — B,*«,*) + 2e, (0,20, — b,2a,") + be, (b,2,— b,0,) = 0. 
So on the right line 6,7, — 6,x, lie three points of intersection of 
the tangents out of V, with the tangents out of O,. We shall indi- 
ll || 
oc 
cate it by A. 
It is evident that these three points and the point OU, are the 
branchpoints for the two collocal series of points in correspondence 
(2,2), determined by the pencils (V,) and (Q,) on the line h. So 
according to a well-known property this (2,2) is involutory. 
Indeed, we find out of 
wt 26,Vuwt+2b,4u74+ 2b,A4n+1=—0 
and 
BiA =De; 
that the (2,2) is indicated on 4 by the symmetric relation 
buu + 2 6, 6, (wu e+ wu?) + 20,5, due’ Hb, = 0 
between the rays projecting it out of O,. 
2. If Q, Q’ is a pair of the involutory (2,2) on h, then the points 
FO 0.0) cand Li (0,Q’, 0,Q) lie-on ¢,,.The line P,P, 
