( 658 ) 
mnt, +k(k+ 1)p=0 
man,” —(k+ 1 z,p=0 : 
me —k’? 2, p = 0 
k being any entire positive number or zero. 
Let us observe, that any four lines having an equation of the 
form (12), give a cross ratio which is independent of the coefficients 
in the equation of /,, or, what is the same, that the cross ratio for 
the same four values of 4 =0,1,2,3,...., is unaltered for all 
the curves of the considered form. 
If we retain the three fixed rays 
CS) hy pl, 
4 MND Pe 
any line of the form (12) gives with these three rays aa absolute 
constant cross ratio for all the curves of the species /:,, where 4 isa 
constant number : 
1 
DC  Ak(kHI) 
also a function of the chosen value of k for the same curve ky. 
6. We have already indicated the two systems of projections, 
the first of which is acquired by projecting the two points of contact 
on the double tangent out of the cusp A, and the second by pro- 
jecting the two common points to 4, and v2, ==0 out of the same 
centre of projection. We take now two of those projections, belong- 
ing to various systems for the same value of 4, having the equations 
Amna, + [(24)° — 1] p= 0 
mne, +-k(k + 1)p=0. 
By the term “same value of #" for the two systems is meant 
that the same number of projections was made in both systems. These 
rays of the pencil | | bear with the fixed pair of rays 
Amna, ES Pp == 0 . . . . . . hd . (d) 
pil 
a cross ratio 4". By means of the parameters 
mn dmn 
k(k-+1)° (2k)?—1" 
we shall obtain 
2k 4 
ed PU Ue eo 
A (an) ( Us ) 
, —Amn, 
