16mia2,* — 4)m?x, x,’ (a?2x, — bx,) + Ix, (aw, — ba,)? — 0, 
and consequently the expression to the left must be divisible by the 
left side of the equation for AD,, AD,, i.e. by 
Ama; — vy (aa, — Bs 
The division gives the equation of the pair 
AE, AE, = Am? — 9a, (a'a, — be) = 0. 
By eliminating w‚(a°w, — br,) out of the latter equation and (4) 
we obtain 
bl 
(me, + nz,2,)? — 4m, = 0, 
therefore 
eek ae ie aw ay 
ADE en) hg a ile (OK 
On the conic (7) are situated the points /,, #,, and on the conic 
8) the points ’,, £’, as the fourth intersections of 4, with the 
pair of lines AL,, AL. 
The equation of ’,4’, will be aequired by combination of (8) 
9 
with (#,); if we eliminate z,°, we obtain: 
! ee 5. es ae == 
FE i, = 4mnz, + 15 (a'a, — bar) = 0 om a; « « (9) 
In pursuing these projections in this manner we can show that 
the general equation of all these lines D',D',, EE’, F’F’,, and 
so on, will be 
Arming. t-\(2k)? Wer br 0 os ee (LO) 
k being any entire positive number or zero. All these projections are 
also elements of the pencil [ /|. 
The parameters in the equation (10) belonging to the mentioned 
projections are of the form 
(2k)?—1 J 
Di = ~—— ,t =k =— 90,1, 2,3,.....). 
Amn 
We conclude from this that 
the cross ratio of any four projections, determined by the 
equation of the form (10) is independent of the coefficients 
in the equation of k,, or, this cross ratio for the same four 
values of k is unaltered for all curves of the considered form. 
The double tangent d, having the equation 
Ane (are Or eas leds, ey» (a) 
belongs also to the projections (10); indeed, the equation (10) fur- 
nishes the equation (d) for k= 0. 
