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O°, breaks up into the two faces through A and into a third plane. 
£ 
Then the curve of contact is plane (see $ 8). 
$ 10. The class of the enveloping cone is equal to the class of 
Oee; the class of Occ, being (P + S)k, as we shall see immediately, 
the class of the enveloping cone also is (P + SS) 4. 
The class of Oj, is equal to its order, the reciprocal polar figure 
of Oe, being also a surface Ozc,. The homogeneous plane coordinates 
(a, 8, y,d) of a tangential plane to O,.., i.e. of an osculating plane 
to a curve C'(p, q, 8) satisfy the conditions (see $ 1, equation 4): 
a pe p i, Y d 
Pe eu PoP Qi bu," Ss cu,sv,% a PQS 
where (7, ¥;,2,) are the coordinates of any point of O,.. and w,, v, 
the parameter values corresponding to the point of contact. By 
replacing 1:u, and 1:v, by u’ and v’ we find: 
u'PvPr 
ig oes 
QSz, 
wav 
B: d= ——_ , 
PSy, 
f u'sy's1 
6 OO = —— . 
POE: 
So, but for constant factors, the coordinates of the pole of the 
tangential plane to O,., with respect to the quadric 
me Hil a i Fe eee 
are equal to the coordinates of a point of Occ, (see $ 4, equation 14). 
So, if the equation of Oz, is 
a Pky Qk 2Sk — B, 
the equation of the reciprocal polar figure with respect to (24) is 
gPk y Qk gSk — —__ KATS ee 
B{PSHQ QP+S SP+Qk 
So the product of the parameters corresponding to two reciprocal 
polar surfaces of the pencil Ove, is constant, viz. 
§PQ+S QSTP SP+Qh—k, 
§ 11. In the case s, =O the asymptotic lines of the system 
C(p:, gs) are right; so according to a known theorem four arbi- 
trary asymptotic curves of the system C(p, g, s) must intersect all 
the generatrices in four points with a constant cross ratio. This 
theorem not only holds for the ruled surfaces on which the curves 
