1367 
On account of p< q<s we have Q<0; so we prefer to trasn- 
form (16) into 
TL ot cake ae Rca ae RT SN ly) 
Corollary I. The degree of the surface O,,, is (P+ Sk. 
For we have: 
PQ HS PQs>0, 
P+S>— 2. 
Corollary II. The surfaces O.-, on which the lines of the systems 
C(p.q,s) and C(p,,q,,5;) are the asymptotic curves form a pencil. 
Corollary HI. The base curve of the pencil of surfaces O,., is 
formed by the sides of the skew quadrilateral OX. YZ 0, each 
of these sides counted a ceriain number of times. 
Corollary [V. The complex of the principal tangents of the pencil 
of surfaces O,,, is formed by the tangents to the curves of both the 
systems C(p,q,s) and C(p,, q,, 5). 
$ 5. Reversely we start from the equation 
. OIL a Sat why MELA LN 
where L, M, N are integers admitting no factor common to all 
three, in order to investigate under which restrictions with respect 
to these numbers the surface represented by (18) admits as asymp- 
totic curves the lines of a system C(p, q,s) and therefore also those 
of the complementary system C(p,, q,,5,). This will be the case if 
the surface (18) contains curves of both systems; to that end we 
must have 
pl +qM +sN=0 
and p,lbt+¢gM+s,N=0, 
or 
(19) 
(ptq+s) (pL+qM+sN) — 2 (pL +qM+sN) =0, 
what can be replaced, on account of (19), by 
Dy ee GaN EEN ATEN ne, (2 
where p, q and s are integers. 
From (19) and (20) we deduce: 
p  —LM+\V{—LMN(L+M+4N) 
oe L(L+N) 
As p and g have to be integers the expression — LMN (L4+M+ WN) 
under the root sign must be positive and a square; so L, M, N 
cannot have the same sign. Let a° be the highest integer square by 
which LAN and 6’ the highest integer square by which L+M+N 
can be divided, so that ZMN:a’? and (L + M+ N):b? contain 
5 89* 
