( 365 ) 



If the coefficients c and h are not zero, if no special relations 

 exist between these coefficients and if besides n, r and m are greater 

 than one, the cyclic point {n, r, m) is equivalent to 

 n — 1 stationary points ^ and to 

 \{n — 1) {n 4- r — 3) + g, - 1| : 2 nodes H. 

 The osculating plane of the curve C in the cyclic point {n, r, m) 

 is equivalent to 



m — 1 stationary planes « and to 

 \{m — 1) (r -f m — 3) + ^, — 1| : 2 double planes G. 

 The tangent of the curve C in the cyclic point {n, r, m) is equi- 

 valent to 



7' — 1 stationary tangents &, to 

 \[r — l){7i -\- r — 3) + ^1 — 1| : 2 double tangents to and to 

 \[r — 1) {r -\- m — 3) -[- ?j — 1| : 2 double generatrices tu' of the 

 developable formed by the tangents of the curve C. 



§ 3. The cyclic point (n, r, m) of the curve C is an tz -j- r-fold 

 point of the developable of which C is the cuspidal curve. 



The cyclic point {n, r, m) counts for 



{n -\- r — 2) [71 + r -j- ^^^) 

 points of intersection of the cuspidal curve C with the second polar 

 surface of for an arbitrary point. 



Through the cyclic point (ji, r, m) of the cuspidal curve C pass 

 \n {n + 2r -f m — 4) + ^3 — g'J : 2 

 branches of tiie nodal curve of the developable 0. 



All these nodal branches touch in the cyclic point {n, r, m) the 

 tangent of the cuspidal curve C (the A'-axis). 



They have with this common tangent in the point of contact 

 |(^ + r) [n + 2r + in — 4.) J^q^-q^\:2 

 points in common. 



The nodal branches passing through the cyclic point {n, r, m) all 

 have in this point as osculating plane the osculating plane ^ = of 

 the cuspidal curve C. 



These nodal branches have with theii' osculating plane z = in 

 the cyclic point {n, r, m) 



|(,^ + ,, 4_ „,) [n + 2?' + 111 - 4) + (^, - q,\ : 2 

 points in common. 



§ 4. The case of an ordinary stationary plane ft, the point of 

 contact of which is a cyclic point (1, 1, 2), shows that through a 



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