VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 495 



x = — cy*- — dy 2 -j- etc. 



If the line xy is a tacnodal edge, having the plane x as tangent plane, we 

 shall obtain two developments, viz. : — 



x = 



= ey 2 + etc., and x — fy 1 + etc. 



If the line is a ramphoidal cuspidal edge, the development will be of the 

 form 



x = 9 V 2 + hy % - + etc. 



In all these cases, a, b, c, d, e, f, g, h depend on the coefficients of the 

 equation of K m . It is evident that, using only the first term in each 

 development, tbese cases reduce to two, viz. : — - 



1°. x = — ay 2 where the sheet of A" ni simply touches the tangent plane 

 along the line. 



2°. x — — cy% where the sheet in question touches the line, and then 

 turns back leaving the tangent plane on the opposite side from that on 

 which it approached it. 



We shall consider the different kinds of lines separately, treating one 

 case of a line of kind I in detail, and giving the results in all other cases. 

 If the line x y is an ordinary line of kind I on M^, the equation of M^ in 

 non-homogeneous coordinates will be of the form 



- o 



(a x x 2 + fii xy -f yiy 2 + etc. ) s + (a 2 x + (3 2 y + y* x 2 +S 2 xy + e 2 y 2 + etc.) = 0. 



Substituting x = — ay 2 in this equation, we get an equation in y and s, 

 which is the equation of a curve that in the neighborhood of xy is the 

 same as the curve of intersection of K m and M^. Of this equation y is a 

 factor, the line xy thus counting once as a line common to K m and M^. 

 From the resultant equation, taking account only of the lowest terms, we 

 obtain 



&2 o. y 2 + a 2 a y — e 2 y — /?« 



s = 



yiy 



Putting y = in this expression, we obtain s = go ; the line xy thus 

 meets the curve above at the vertex, i. e. the curve of intersection of K m 

 and M^ passes through the vertex. (Similarly if x = — cy$, there is a 

 point of the curve at the vertex.) If, however, /3 2 = 0, i. e. if the cone 

 and the monoid touch along xy, we get a point of the curve distinct from 

 the vertex. For y factors out once more from the above, that is the 

 line xy counts twice as a line common to K m and M^ and we have 



