( 182 ) 



plane determined by the A'-axis and the point where C" touches x = 0. 

 When A = the terms of the lowest order in t in the coelTicients 

 «0, a^ and a^ are respectively of order 2, 2 and 0. 



The terms of the lowest order in t of the discriminant appear in 

 the first term of the equation {E) at the end of the preceding §, 

 namely in the tei-m 1na/i^(p^. So the terms of the lowest order 

 in t are 



Ca (3;> + Ihp') f 



where C represents a constant. The discriminant possesses three roots 

 ^ = or the A-axis counts for three common tangents of 6" and S> if 



a (3j9 + 46p^) = 

 or if 



a 1= 0, 3 + 46p = 0, 2^ = 0. 



If 9 -f- -^J)]} = 0, the surfaces D" and S have according to (C) a 

 stationary contact, as A is also equal to nought. The origin Pis now 

 an ordinary point (not a parabolic or double point) on the surface 

 S and the common tangent (the A"-axis) does not coincide with one 

 of the inflexional tangents of S in P. 



This furnishes the theorem : 



If an arbitrary surface S and a devehpahJe D^ have a stationary 

 contact in an ordinary point P of both surfaces and the generating 

 line I of D^ through P is neither of the two injieicional tangents of 

 S in P, then I counts for tJiree common tangents of the cuspidal 

 curve 6" and of S. 



If a = the surfaces D^ and ;S' have according to (C) still a 

 stationary contact, as still h = 0. The origin Pis now a parabolic point 

 of ;S' whilst the A-axis is the only inflexional tangent. The coefticienls 

 ag, a^ and a^ all contain the factor f . So the discriminant possesses 

 the factoi' t\ so that now the discriminant has four roots / = 0. So 

 the X-axis now counts for four common tangents of C^ and S. 



If 2) = 0, then C^ touches S in the origin P, whilst the osculating 

 plane of C^ in P coincides with the tangent plane of ^S' in P. 

 The terms of the lowest order in t in the coefiicients ag, a^ and a^ 

 are now respectively of order 3, 2 and 0. So the discriminant (P) 

 is divisible by f, so that C' and S now have in the origin 7^ three 

 common tangents. Writing in the equation (7i) of the surface S for 

 the coordinates of a point on C' the expressions ,r = /,?/ = f , c = /^ 

 we obtain an equation in t, containing f^ as a factor. The curve C^ 

 has thus in the origin only two points, but three tangents in common 

 with aS'. 



If h = a =: p =zO, then 6"' touches the surface S in a parabolic 



