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cyclic point branches of the nodal curve can pass not touching in 

 this point the cuspidal curve. 



These intersecting nodal branches exist only when 5-3 ]> 1. If 

 r ^ 1 the coefficients b and c must satisfy special conditions. 

 If r =zl then through the cyclic point {n, r, m) of the cuspidal 

 curve pass eilher 5', : 2, or {q^ — 1) : 2 of these nodal intersecting 

 branches. All intersecting nodal branches have a common tangent 

 in the plane ^ = if r = 1. 



§ 5. The case of an ordinary stationary point ^ (2,1,1) shows 

 that through a cyclic point of the cuspidal curve nodal branches 

 can pass which have the same tangent, but not the same osculating 

 plane as the cuspidal curve. These particular nodal branches exist 

 only when ^'^^l. lï q^'^1 and m = l these particular nodal 

 branches are always present. \i q^'^1 and also m ]> 1 the coefficients 

 h and c must satisfy special conditions. These particular nodal 

 branches have in the cyclic point {n, v, in) a common osculating 

 plane (differing from the plane 2 = 0) if m = 1. 



§ 6. The tangent to C in the cyclic point {n, r, m) is an r-fold 

 generatrix y on the developable 0. The r sheets of the surface 

 passing through the generatrix g all touch the osculating plane z = 

 of C in the point {n, r, m). 



The generatrix g moreover meets in q — {n -{- 2 r -\- m) points R 

 a sheet of the surface 0, when is of order q. 



In every point R the generatrix g meets r branches of the nodal 

 curve. These ?" branches form, when m'^ r a singularity {r, r, rn — r) 

 and the osculating plane of these nodal branches is the tangent 

 plane of along g. 



If m <^ 7' these r nodal branches form a singularity (r, in, r — m) 

 and the osculating plane of these r nodal branches is the tangent 

 plane of along the generatrix intersecting g in R. 



If r = m these r nodal branches form a singularity {r, r, 1). 



§ 7. In general the singular generatrix g will meet only nodal 

 branches in the cyclic point {n, r, m) and in the points R. If ^'^ ^ 1 

 the generatrix g may meet moreover nodal branches arising from 

 the fact that some of the r sheets, which touch each other along g pene- 

 trate each other. These nodal branches meet g in the same point Q. 

 If g'j > 1 and ?? = 1 there is always such a point of intersection Q. 

 \i q^'^l and n > 1 the coefficients b and c must satisfy some special 

 conditions if the sheets passing through g are to peneti'ate each other. 



