49-4 PROCEEDINGS OF THE AMERICAN ACADEMY. 



lines of M^. The case where w^j^o = = '^'^k-\-g—\ = 0, that is 



where the line is a (k + ^)-tuple line on the superior cone while only a 

 I'-tuple line on the inferior cone belongs to this case. We assume 

 3 ^ ^ ^ iu — k. The line now counts for k (k + g) lines of M^, other- 

 wise it is not different from the general line belonging to this case. 



We thus have in reality four kinds of lines, as shown by the four cases 

 above. We shall designate these lines as of kinds I, II, III, and IV, 

 respectively. Lines of kinds I, III, and IV are torsal on all sheets of 

 Mfi that contain them, whereas lines of kind II are scrolar on at least 

 one sheet of the monoid. Lines of kind III differ from lines of kind 

 IV only in regard to the breaking up of the tangent cone at the 

 {k + l)-tuple point. 



3. Consider now the character of the curve of intersection of Km and 

 Jil^ in the neighborhood of lines of the above kinds when they are 

 common lines of Km and iH/^. The point where the line is crossed by a 

 branch of the curve can be determined by considering the intersection of 

 J/^ by the tangent plane to a sheet of Km along that line. AVe can, 

 however, determine more than this. If a plane, say a: = 0, is taken to 

 be the tangent plane to a sheet of the cone along the line xi/, we can 

 develop x in terms of y.* We neglect all terms in the development 

 after the first, as we are concerned only with points in the immediate 

 neighborhood of the line x y. Substituting the value of x from this 

 development in the equation of the monoid that has the line xy as a line 

 of one of the above four kinds, we obtain not only the point of crossing 

 of the line by the curve but also the direction of the tangent to this curve 

 at that point. The equation of the cone Km having the line xy as a. 

 ^•-tuple line can be put into the form 



t,,zm-k + ;;t+i 2"'-^-l + + t,n-l z+tm = 0, 



where /« is a homogeneous function of x and y of degree a. This equa- 

 tion can be transformed into an equation in x and y alone (putting 

 2 = 1). If the plane a; is a tangent plane to an ordinary sheet of 

 Km through the line x y, the development will be of the form 



X = — ay"^ + by^ + etc. 



If the plane a: is a tangent plane along a keratoidal cuspidal edge of K^i 

 we shall have 



* This can be done by the Newton-Cramer method, explained in Salmon's 

 Higher Plane Curves (1873), p. 44. 



