496 PROCEEDINGS OF THE AMERICAN ACADEMY. 



8 2 ay + (« 2 a — *z) 



s = 



a — e 2 



yi 



This reduces to s = — for y — 0. The curve of intersection thus 



yi 



crosses xy at the point ( x = y = 0, s = 1. Transforming our 



coordinates so that a 2 a = e 2 , we obtain (x = y = s = 0) as the point of 

 crossing and yi s — 8 2 «^ = 0, x = as the equations of the tangent line 

 to the curve at that point. Thus it is evident that in the case of lines of 

 kind I, whatever be the multiplicity of xy on J/ M , if no sheet of K m 

 touches a sheet of M^ along this line, the curve of intersection will 

 always pass through the vertex. As we are considering a point off the 

 curve as vertex such cases need not be considered. In the case where 

 two sheets of K m unite to form a cuspidal sheet that touches a sheet on 

 Mp. along xy, the curve of intersection will either pass through the 

 vertex or it will have x y as tangent line at a point of it. As we have 

 also assumed the vertex not to lie on a tangent to the curve, these cases 

 may both be avoided. There will be a point of the curve on xy distinct 

 from the vertex and not having xy as tangent at it, only in the case 

 where a sheet of K m touches a sheet of M^ along the line xy and the 

 sheet of K m is one for which development 1° on page 35 holds. It does 

 not matter whether the sheet of M^ is single, cuspidal, or tacuodal. 

 There will be as many branches of the curve crossing xy as there are 

 such sheets of K m touching sheets of M^. As the vertex has been taken 

 in such a way that the curve has no apparent multiple point of multiplicity 

 greater than two, the case where there are more than two distinct points 

 of the curve on xy may be avoided. For special relations between the 

 coefficients, k of the points on x y may coincide and form a K-tuple point 

 on the curve. This will, however, in general necessitate the monoid to 

 be of an order higher than m — 1. For the equation of il/ OT _ i that has 

 a line as a K-tuple line of kind I contains in general m 2 — k 2 — 2 k — 2 

 arbitrary constants. In order to make this monoid contain a curve C m 

 that has a point of the K-tuple line as a K-tuple point we must make it 

 contain m (m — 1) — k 2 -f- 1 additional points of C m . We can therefore 

 always make M^ cut C m out of K m , if 



m ( m _ 1) _ K 2 + 1 = m 2 _ K 2 _ 2 K _ 2, 



i. e. if 2 k + 3 < m. 



Thus if k = 3, it is only possible when 9 ^ m; if k = 4, when 11 < m; 

 etc. As will be shown later, the curve C m that has a multiple point can 



