TRANSACTIONS OF SECTION A. oil 



express the relation between the discriminant and the surface or curve A = near 

 the origin. 



(1) In general the shape of the discriminant is given by 



= A, + A,-iB,-7Co 



showing that the discriminant and the surface A have a common tangent plalld 

 A, = 0; further if Aj = both surfaces have conical points and the tangent cones 

 touch along two generators. The same algebra when interpreted in two dimen^ 

 sions proves that the discriminant is an envelope, and that if it happens that the 

 enveloped curve has a node at its point of intersection with the envelope, then the 

 latter also has a node with a different pair of tangents in general. 



(2) If Cjj = the discriminant is 



= A.U2A,P, + .,\B^^/D,+ .. . 



showing that in space the curve -^ = 0, \//'( = 0, \/',( = 0, is a cuspidal edge, and Iti 

 the plane these points are cusps on the discriminant. Similarly it may be proved 

 that the discriminant surface has a nodal line and the discriminant curve isolated 

 nodes. 



(3) If Bj = XA, the leading terms in the discriminant are 



A, + A. + A3 + A, - J- (B,-XA,)-/C„ 

 which indicates that, in the plane, at isolated points of the envelope given by 



the contact is of the third order, with a corresponding result lU space. 



(4) If C(, = and Do = 0, which happen at isolated points of the discriminant 

 surface given by 



x/. = 0,V.( = 0,^//;( = 0, ./,,„ = 0, 



three sheets of the surface have a common tangent plane, and the point of contact 

 is a stationary point on the cuspidal edge, lying also on the nodal line. 



Necessary and sufficient conditions that the envelope of a family of plane 

 curves may osculate one member are found to be Bi = \Ai, Co = 0; or, in other 

 words, that it should be possible to find .i-, y, t to satisfy 



^ = 0, V'« - 0, ^\t, i|r,„ = ^/.J, ^,„ rj^n = 0, 



and then the discriminant has two branches, cacli osculating the cul've at this 

 point. The case in which one of these branches is the enveloped curve is 

 discussed. 



AVhcn the conditions 



are satisfied at all points of a curve, this curve is an osculating envelope and occurs 

 as a squared factor of the discriminant. In the case of the family of circles ot 

 curvature of a curve the discriminant contains as factors all the superoscnlatino- 

 circles and the point circles at the foci and the curve itself as a squared factor. 



These theorems in connection with plane curves are discussed also in a "■eo- 

 metrical manner. One method consists in taking the parameter as a third 

 coordinate and projecting the surface so obtained by rays parallel to the third 

 axis. Another method is to regard a curve as a succession of near but isolated 

 points, and in this way some easy intuitive proofs are given, which require, how- 

 ever, to be supplemented by more rigorous investigations. Finally, the analytical 

 method is applied to obtain the usual results in connection with node- and 

 cusp-loci. 



