( 657 ) 
e- the line AB =O cuts A, stil? in two points P,, P,; 
projecting these points out of the cusp A, we obtain two lines having 
as equation 
ONO ea Ue a hs CED 
Eliminating wv, (a°«, — be,) or #,.p out of &, and the latter equa- 
tion, we have 
(ma, + nae) + mz,’ — 0 
therefore 
c=) DAO, en se doce 
ONG a) woe ee AD Baten wer? 2). (D) 
The equation (/) defines a conic, passing through the two points 
PP’, in which the curve &, is cut still by the pair of lines (11). 
By eliminating «,’ out of (4,) and (6) we obtain the equation of 
Er. in the form: 
PP, =mne«, + 2p = 0. 
On the line P/P’, lie two other points Q,,Q, common to /:, and 
PP’; so we can now project the points Q,,Q, out of A by two 
lines cutting 4, still in the fourth intersections Q’,,Q’,, and so on. 
There is no difficulty to show, that the general equation of all 
Bies projections PP QS 9 eneen Will he 
mind, lik pd; (Ol ey arn ie se (12) 
and we see, that all these projections are again elements of the 
pencil [A]. 
By means of the invotutory pencil of conics (1) we find with 
respect to (3) and (12): 
k(kH1) u 
mn (2—— mu)? 
From this equation follows: 
n (k-+1) : nk 
rn 
mk m (k+1) 
Ub, 
therefore any line having the form (12) cuts #, on the two conies: 
m ka? + n(k + 1) ¢,¢, = 0 
(4 = 0,1,2,8,. ) . (13) 
m(k+1)e2,? + nk we, =0 
By eliminating 7, out of (12) and (13) we obtain two pairs of 
pencils with non-consecutive rays in correspondence (1,2), by means 
of which any number of discrete points of 4, can be determined ; 
thus 
