1366 
can immediately be verified. Likewise one finds 
PP, +19, + 8,5, = 0 
and as according to (41) P, =— P, ete. we find 
pP+¢Q+3S=9, 
and also NEEN A a aoa 
PrP Ha Qd s,S=0. 
§ 4. Let 0, be a surface determined by the equations 
“c= auPvPi, 
y = buiva, EERE Der 
z= cue vi, 
where the coordinate lines wv = constant are curves of the system 
C(p‚g,s) and the coordinate lines «= constant curves of the com- 
plementary system C'(p,,4q,,8,). The two coordinate lines passing 
through any point P, (#,, y,,2,) Of Ove, admit in this point the same 
osculating plane. This common osculating plane contains the tangents 
in P, to both the coordinate lines and as the director cosines of 
these tangents are proportional to 
REEN 
and Prana Sr San 
these tangents do not coincide and the eommon osculating plane is 
at the same time the tangent plane of QO... in P,. 
This proves the theorem : | 
The two systems of coordinate lines are the systems of asymptotic 
curves of the surface Orc, gwen by (14). 
In any point P, of O.., the tangents to Cp, (p‚q,5) and Cp,(p,.91,51) 
are the principal tangents as these curves are the asymptotic curves. 
So in any real point of Q,,, the principal tangents (see (15)) are 
real and different from one another; so we have the theorem: 
All the points of Occ, are hyperbolic. 
The equation of the surface O.., is: 
(=) (2) (=) = jes oe ae ae 
k being the lowest common multiple of the numerators P, Q, iS of 
(3) after reduction of these fractions to their simplest values. In- 
deed the values (14) of the coordinates of any point of O,., satisfy 
the equation (16) for arbitrary values of w and v, as according to 
(12) and (18) we have the identities 
pP-+qQ+s8=—), 
PP+U9+ 3, S= 9, 
