1369 
pho ee, xt tp ’ y = y, ttn, a zien 
as from the identities (12) and (18) we can deduce 
P(p+Ap,) + Q(q+4q.) + S(s+As,) = 0. 
If A, and A, represent any two definite values of 4, the cross ratio 
of the four tangents in a point P, of O.., to the curves through P, 
of the systems 
C (Psd 5), C (Pir dis 8s) EO (PAP ada sos), 
C (p FA, Pi +491 s+A,3,), 
is always equal to 2, :2, and therefore independent of «,, y,,2,. So 
this cross ratio is constant all over the surface. 
If we put e.g. 
: 4,=1:(p+q-+s) 
A, = (prate): pts) (pats) (P+9—3) 
the two systems of curves corresponding to these two values of 2 
are the systems C(p?; q’, s?) and C(p,’, q,’, 5,°). 
So the cross ratio of the tangents in any point of O,., to the four 
curves through this point belonging to the systems 
C(p‚q,s), C(pyauns) (ps 97, 8"), (175 9175 817)» 
is therefore 
_ ptt) (p—ats)(pt9q—3) 
1 (tat (p, tate) 
This cross ratio becomes zero or infinite if two of the four tangents 
coincide with each other; then the curves touching these coinciding 
lines also coincide. For p, q, and s positive and p<q<s the cross 
ratio becomes zero under the condition p + g==s only and infinite 
for p,+q,-+s, only. In the first case the systems C'(p,,q,, s,) and 
C(p,7,91, 5,7) coincide, in the second case the systems C(p,q, s) and 
C'(py73917551")- 
Reversely, if a curve of a system C’'(p’',q',s') lies on the surface 
Occ, it will always be possible to find a value of A for which the 
system C(p + 4p,,q + 4q,,5 + 4s,) coincides with the system C'(p', q', s’). 
So we have to prove that it is possible to tind values 4 and u 
satistying the three equations: 
pt 4p, + pp —=0, 
p + âg, + ug =0, 
s+ ds, + us = 0. 
These three equations are not mutually independent; multi- 
plying the first members by P, Q and S and adding the results 
res | 
1° 
=P1915:2P98 (pta Hs) (pt+qi+s,)}- 
