( 54 ) 
system of the lines g intersecting all the lines (4) by (9), there exists 
between the two incident systems of lines (y), (d at infinite distance 
this reciprocal connection that the plane passing through an arbitrarily 
chosen point QO at finite distance and an arbitrary line of one of the 
two systems is a plane of position for the pair of the planes connect- 
ing the point OV with two arbitrary lines of the other system. From 
the manner in which the quadratic scroll (/) is formed out of g,, 
Yo J.» Jy Namely ensues that the surface //° of order two, bearer 
of the two systems (y), (4, agrees with itself in the polar system 
at infinity in S, of which the imaginary sphere 22 common 
to all hyperspheres is at the same time the locus of the points lying 
in their polar planes and the envelope of the planes passing through 
their poles; for to each line ¢ intersecting y,, g,, 9,’, g’, regarded as 
locus of points agrees in that correspondence a line ¢’ lying with 
Yi > Jo's Yrs Jo M a plane regarded as axis of a pencil of planes, ete. 
If we make the lines y, g’ of (g) normally conjugate to each other 
to correspond to each other, then between the lines of (9) a quadratic 
involution is formed; the double rays g;,g; of this involution must 
lie on Ay, being normally conjugate to themselves. Likewise does 
45; contain the double rays ¢,, t of the involution formed in quite 
the same way between the lines f, {normally conjugate to each 
other. So H and 42 intersect each other in the sides of a skew 
quadrilateral whose pairs of opposite sides are the double rays 
(gi, g,) and (ty, t) of the pairs of rays (g, g’) and (¢,t’) of (g) and (#) 
normally conjugate to each other. 
If (g,g’) are two normally conjugate rays g and (f, t’) likewise 
two normally conjugate rays ¢ and if we take the planes Og, Og’ 
and Ot, Ot’ as coordinate planes O.X,X,, OX,X, and OX,X,, OX,X,, 
then the surface M/ must be characterised with respect to this rectan- 
gular system of coordinates by the equation er, = wr, between the 
infinite coordinates. For the quadratic surface vr, + priv, = 0, 
brought through the lines at infinite distance of the four mentioned 
coordinate planes, corresponds to itself in the polar system with the 
sphere wv? + 2,? + .,? +.7,—=0 at infinite distance as surface of 
incidence only when the absolute value of p is equal to unity. For 
the normally conjugate Jine of «, = 2x,, Av, + pr,=0 is Ar, +2,=0, 
pe, =de, and this new line lies only for p?—=1 with the original 
one on «, 2, + Pt, ‚== 0. So by reversing if need be the sign of 
one of the coordinates we can always bring the equation of HZ into 
the form Sn een wae 
3. Before passing on to the general case we shall consider the 
